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បជុំវ គណិតវទសិសពូែកគណិតវទ ក់ទី១០ បកែបេយ កវ សិរ" គណិតវទអូំពិកបៃពណីេវៀតម 30-4 ំ 2011

Vnmo 30 4-2011-grade 10

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Page 1: Vnmo 30 4-2011-grade 10

�បជំុវ�� គណិតវ�ទ�សិស�ពូែកគណិតវ�ទ��� ក់ទី១០

បកែ�បេ�យ ែកវ សិរ"

គណិតវ�ទ�អូ�ំពិក�បៃពណីេវៀត�ម 30-4 ��ំ 2011

Page 2: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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វ��� �អូពំិក�បៃពណីេវ�ត�មេល�កទី XVII �� ំ២០១១

មុខវ�%& ៈ គណិតវ�ទ)*� ក់ទ ី១០ រយៈេពល ១៨០/ទី

�. �����យម� ��ង�� មក��ង�ន��ច�ន�នព��� 3 2 315 78 141 5 2 9x xx x− + − = − �

�. �គ !ច�ន�នគ�"# �ជ%&ន nន�ង 1 2 3 4d dd d< < < 'ប�ន��)ចកគ�"# �ជ%&ន�*ចប�ផ���ប" n�

�ក�គប"ប,- ច�ន�នគ�"# �ជ%&ន n�./ម0� ! 2 2 2 21 2 3 4n d d d d+ + += �

1. �2ក��ងប3ង"�គ !ម�� xOy ន�ងព��ច�ន�ច A4���2�5/កន3�ប67 �" Ox ន�ង B 4���2�5/កន3�

ប67 �" Oy 89ង, !���� : OAB ម;���ង" O � ∆ 'ប67 �"ច5<�ម�យម�ន �"

=មO , )�?)�ង �"=មច�ន�ចក,- 5 I �ប" AB ន�ង �"កន3�ប67 �" ,Ox Oy ��@ងA�

��ង"ប,- ច�ន�ច ,C D � =ង M 'ច�ន�ចក,- 5�ប" ,CD N 'ច�ន��បពBCន OM នDង ,AB H 'ច��,5)កង�ប" N �2�5/ CD � �ព5 ∆ ច5<�, ច*��ក�ន��ច�ន�ច�ប"

ច�ន�ច H �

E. �គ ! ,,a b c 'ប�ច�ន�នព��ម�នF# �ជ%&ន�ផ7GងH7 �" 2 2 24 9 14a b c+ + = �

��យបI% ក"J� 3 128b c abc ≤+ + �

K. ��យបI% ក"Jព� 2011ច�ន�នគ�"# �ជ%&ន,កL��យ �គ)�ង��ជ/�� /;នច�ន�នព��

).5ផ5ប*ក� Mផ5.ក�ប"?)ចក�ច"នDង 4018�

N. �គ !�F5�ប ( )2 2

: 18 4

Ex y+ = ន�ងប67 �" ( ) : 2 2 4 0x y∆ − + = � =ង ,B C ��@ង

A� 'ច�ន�ច�បពB�ប" ( )∆ ន�ង ( ) , B CE y y> �O/យ A'ច�ន�ច�2�5/ ( )E 89ង

, !�ប)#ងព� A �P ( )∆ )#ងប�ផ��� �កច�ន�ច M �2�5/ ( )E �./ម0� !�ប)#ងព� M

�Pប67 �" AB គM)#ងប�ផ��� '()&'()&'()&'()&

Page 3: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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ចំេល�យ

�. ម� �).5 !មម*5នDង�

( )3 33 5) 925 9( 5 x xx = − + −− + −

=ង 3 25 9y x+ −= , �យ/ង;ន�បព<នQម� ��

3

3

3 5 9

( 5)

( )

9

5

2

x y

y x

x = − + −− = −−

.កFងRនDងFងRម� � (1) ន�ង (2) , �យ/ង;ន�

3 3( 5( 5 5 5) )x y x y− − − = − + .

2 2( ) ( 5 ( 5)( 5) ( ) 5) 05x yx y yx⇔ + − − + − + − − =

��យ 2 2( 5)( 5) ( 55 ) 5( ) x y yx + − + −− − + =

2

21 35 ( 5) 5 0, ,( 5)

2 4xy yx y+ = − + − −

+ > ∀

∈ℝ

�6�ម� � (3)មម*5នDង x y=

ព� (2)Sញ;ន 3 3 22 9 15 73 1) 1 6( 5 0x xx x x= − ⇔ − + − =−

2 11 29)

4

( 4)( 1 50 1

2

x

x

x xx

⇔ − + ==

⇔ ±− =

.*ច�ន� �ន��U�ប"ម� �).5 !គM 11 5 11 54; ;

2 2S

− + =

�. �ឃ/ញJ 2 0 (mod 4)x ≡ �ព5 x គ*, 2 (m1 4)odx ≡ �ព5 x ��

�ប/ n 'ច�ន�ន� �6��គប"ប,- ច�ន�ន id �ទQ)���O/យ

2 2 2 21 2 3 4 1 (mod1 1 1 0 4)dn dd d≡ + + + ≡ + + + ≡ (ក�:� �ន�ផ7�យព� �ព��)

.*ច�ន� �យ/ង;ន 2n k=

�ប/ 4'��)ចក�ប" n �6� 1 1d = ន�ង 2 22 3 41 02,nd d d+≡ ++= )ចកម�ន�ច"នDង 4 (ក�:�

�ន�ផ7�យព� �ព��)� .*ច�ន� �យ/ង;ន n )ចកម�ន�ច"នDង 4�

.*ច�ន� { } { }1 2 3 4, , , 21 ,, ,d d d p qd =

Page 4: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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� M { } { }1 2 3 4 1,, , , 2, , 2d pd d d p= ច��Z� ,p q 'ប,- ច�ន�នប[ម�

ក��ងក�:� { } { }1 2 3 4, , , 21 ,, ,d d d p qd = �យ/ង;ន (m3 4)odn ≡ (ផ7�យព� �ព��)

.*ច�ន� ( )25 1n p= + �O/យ n )ចក�ច"នDង 5 , �6� 3 5p d= = ន�ង 130n = �

1. • ��យបI% ក")ផ�ក�ប�

=ង�

P+ 'ច�ន�ច�បពB�ប"កន3�ប67 �" OI

នDង�ងBង" ( )C \�Dក��]���� : OCD �

J+ 'ច�ន�ច�បពB�ប"កន3�ប67 �" OI

នDងប67 �")កងនDងកន3�ប67 �" Ox ��ង" A �

,E F+ ��@ងA� 'ច��,5)កង�ប" P

�P�5/កន3�ប67 �" ,Ox Oy �

=មប�^ប"�យ/ង;ន�

� �ប/ C A≡ , 6� ! D B≡ (� Mផ7�យមក# �ញ) �6� M N H I≡ ≡ ≡ �

� �ប/ C A≠ ន�ង D B≠ , �ព5�6�,

+ កន3�ប67 �" OI 'ប67 �"ព���ប"ម�� �AOB P⇒ 'ច�ន�ចក,- 5ធ�* �CD�ប" ( )C

CPM D⇒ ⊥ � ព��6�, , ,E M F 4���2�5/ប67 �" Simson �ប"ច�ន�ច P ច��Z� OCD∆

+ កL��Z� កន3�ប67 �" OI 'ប67 �"ព���ប" �AOB �6� ||EF E BI FO A⊥ ⇒

,AJ EP+ )កង��មA� �PនDង Ox �6� ||AJ EP

ព��6� =ម�ទD-�បទ=�5 �យ/ង;ន�

|| , ,OJ OA ON

NJ PM NJ N H JOP OE OM

CD⇒ ⇒ ⊥ ⇒= = ��"��ង"ជ���

.*ច�ន� H 4���2�5/�ងBង" ( )T Fង̀�"ផa�� IJ � ��យប,- កន3�ប67 �" ,Ox OI ន�ងប,-

ច�ន�ច ,A B �2នDង�6� ,I J �2នDង ( )T⇒ �2នDង�

• 5�ម���ប"�ន��ច�ន�ច�

=ង 1 1,C D ��@ងA� 'ប,- ច�ន�ច4���2�5/ ,Ox Oy 89ង, ! 1 ||IC Oy ន�ង 1 || ;ID Ox

1 2,H H ��@ងA� 'ច�ន�ច�បពB�ប" ( )T នDង 1IC ន�ង 1ID � �ព5�6� C bច4���2Fង̀�"

N

B

A

O

1C

E

F

y

x P

2H

1H

1D

D

H

I

Page 5: Vnmo 30 4-2011-grade 10

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1OC )�ប9��,c � .*ចA� នDង D bច4���2��]Fង̀�" 1OD )�ប9��,c �� Sញ;ន H 4���2

�5/ធ�* �1 2H H &នផ7�ក I �ប"�ងBង" ( )T (�5/ក)5ងព��ច�ន�ច 1 2,H H )�

• ��យបI% ក")ផ�កផ7�យ�

�2�5/ធ�* �1 2H H &នផ7�ក I �ប"�ងBង" ( )T �យ/ង�dច�ន�ច 'H � =ង ', 'C D 'ច�ន�ច�បពB

�ប" 'IH ��@ងA� �PនDង ; ',Ox Oy P 'ច�ន�ច�បពB�ប"កន3�ប67 �" OI �PនDង�ងBង"\� Dក

��] ' '; ;OC D E F′ ′∆ ��@ងA� 'ច��,5)កង�ប" 'P �P�5/ , ;Ox Oy N ′'ច�ន�ច�បពB

�ប" 'JH �PនDង ;AB M ′ 'ច�ន�ចក,- 5�ប" ' 'C D � �យ/ង��e# �បfg ញJ , ',O N 'M

��"��ង"ជ��A� �

ព��'.*ច�ន� =ង ''N 'ច�ន�ច�បពB�ប" 'OM ន�ង AB � �ព5�6� =ម�^ប"បI% ក"

)ផ�ក�ប �យ/ង;ន�

' '''' || ' '

' ' '

OJ OA ONN J P M

OP OE OM⇒= =

' '' ' ,' ',C D H JN J N ′⇒ ⊥ ⇒ ��"��ង"ជ�� ' '' NN⇒ ≡

.*ច�ន� , ', 'O N M ��"��ង"ជ��A� (បIg ��e#��យបI% ក")�

• ន���h ន� �ន��ច�ន�ច H 'ធ�* �1 2H H &នផ7�កច�ន�ច I �ប"�ងBង" ( )T (�5/ក)5ងព��

ច�ន�ច 1 2,H H ) �

E. # �មiព).5 !មម*5នDង� 24 )16 (16 2b c abc ≤+ +

+ Fន�#�-នj# �មiព Cauchy �យ/ង;ន�

2 2 2 23( 1) 8( 1)6 116 3 81 b c bb cc ≤ + + + = ++ + .

2 2 2 2 2 2 2 2 24 9 )11 ( 25b c a b c a ba c+ + − − − − −+ = −= .

�6� �./ម0���យបI% ក" (1) �យ/ង�Aន")���e# ���យបI% ក"# �មiព�

2 2 2 1 2 (2)b c abca + + − ≥ , ច��Z� ,,a b c ម�នF# �ជ%&ន�

+ =មប�^ប" 2 2 24 9 14a b c+ + = , # �មiព (2)bច����k/ង# �ញ.*ច�ង�� ម�

2 2 2 2 2 2) ( 4 94 28( )1 b c b c abca a+ + − + + ≥ .

2 2 210 51 23 8b c abca⇔ + + ≥ .

2 2 2 2 2 210 5 ) 4 9 28(13 14b c ba bca a c⇔ + + + + ≥

Page 6: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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+ Fន�#�-នj# �មiព Cauchy ម-ង�ទ@� ច��Z� 28 ច�ន�ន�យ/ង;ន�

2 2 2 2 13 2 10 2 5 13 10 5142810 5 28 ( ) ( ) (3 ) 281 b c a b a b ca c+ + ≥ =

ន�ង ( )22 2 2 2 2 4 2 9 4 914144 9 14 ( ) ( ) 14b c a b c aba c+ + ≥ =

+ .*ច�6� ( )22 2 2 2 2 2 13 10 5 4 914 1410 5 ) 4(13 14 228 89 14b c b c aa a abcb c ab c =+ + + + ≥

មiព�ក/�&ន�ព5 ( , , ) (1,1,1)a b c = �

5�l�"��e#;ន��យបI% ក"��ច^5"�

K. �ព5)ចកច�ន�នគ�"# �ជ%&ន,ម�យ នDង 4018 �6�ប,- �:5"��e#4���2ក��ង�ន��

{ }1, ...0, , 4017 �

ក��ងប,- �:5"�ង�5/ �យ/ង)ចក�ចញ'�កmម.*ច�ង�� ម�

+ �កmមទ�ម�យ ��ម&នប,- ច�ន�ន�ព5)ចកនDង 4018 ;ន�:5"�n/ 0

+ �កmមទ�ព�� ��ម&នប,- ច�ន�ន�ព5)ចកនDង 4018 ;ន�:5"�n/ 1 � M�n/ 4017

+ �កmមទ�ប� ��ម&នប,- ច�ន�ន�ព5)ចកនDង 4018 ;ន�:5"�n/ 2 � M�n/ 4016

............

+ �កmមទ� 2009 ��ម&នប,- ច�ន�ន�ព5)ចកនDង 4018;ន�:5"�n/ 2008 � M�n/ 2010

+ �កmមទ� 2010 ��ម&នប,- ច�ន�ន�ព5)ចកនDង 4018 ;ន�:5"�n/ 2009�

.*ច�ន� &នS�ងF" 2010�កmម, )�)ប�'&ន 2011ច�ន�ន �6�=ម�ទD-�បទ Dirichlet

�?ងព�ក? ��e#&នព��ច�ន�ន ).5�:5"ក��ង�ប&:#�ធ�)ចកនDង 4018 o3 ក"ច*5ក��ង�កmម

'ម�យA� �

�ន� គM'ព��ច�ន�ន).5��e#�ក ��Z��ប/ព��ច�ន�ន�ន� &ន�:5"�n/A� �6�ផ5ង�ប"

ព�ក?)ចក�ច"នDង 4018, �ប/ព�ក?&ន�:5"�ផpងA� �6�ផ5ប*ក�ប"ព�ក?)ចក�ច"

នDង 4018 �

N. =ង [ ]2cos ) ( )(2 2 si ;2n 0; ,t E tA t π∈ ∈

�យ/ង&ន 4 2 sin 4

4sin 4cos 4 4( , ))(

6 6

tt t

d A

π − + − =∆+

=

Page 7: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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.*ច�ន� )( ); (d A ∆ ធ�ប�ផ���ព5

3sin( ) 1 (2; 2)

4 4t t A

π π⇒= = −⇒−

0 (0;2)

1( ;( sin 3

4 2;)) 0

( 2 0)22

t Bd A t

t C

ππ⇒

∆ = ⇒ ⇒

= − = − = ⇒

ម� � ( )( ) 2 4: 2 2 0AB x y+ + − =

( ) 8 2 sin cos 42 2 2 2 sin 4cos 4 8 8( , ( ))

10 4 2 10 4 2

tt td M AB

π π + − + + − = =+ +

.*ច�ន� ( , ( ))d M AB )#ងប�ផ���ព5

32c

11 3sin( ) 1 ( 2 2 sin ;

8 8)

88ost Mt

π π π π⇒ −== − ⇒ −+ �

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១

�. �����យម� ��ង�� ម�5/�ន��ច�ន�នព��� ( )3 28 8 (19 2 )x x +=+

�. បfg ញJ ម�ន&នប,- ច�ន�នគ�" ,,x y z �ផ7GងH7 �"ទ�6ក"ទ�នង� 2 2( 2012) ( 2008)( 2014)( 2010)x x x y z y z x+ + = + + + + − −+

1. �គ !���� : ABC \�Dកក��ង�ងBង" ( )O ន�ង\� Dក��]�ងBង" ( )I � =ង , ,D E F ��@ង

A� 'ប,- ច�ន�ចប9��ប" ( )I �PនDងប,- �ជmង , ,BC CA AB � ង"�ងBង" 1( )O ប9�

��]នDង ( )I ��ង"ច�ន�ច D �O/យប9�ក��ងនDង ( )O ��ង"ច�ន�ច K , �ងBង" 2( )O ប9���]នDង

( )I ��ង"ច�ន�ច E �O/យប9���]នDង ( )O ��ង" M , �ងBង" 3( )O ប9���]នDង ( )I ��ង" F

�O/យប9�ក��ងនDង ( )O ��ង"ច�ន�ច N � បfg ញJ�

).a ប,- ប67 �" , ,DK EM FN �"A� ��ង"ច�ន�ច P �

).b ប67 �" OP �"=មF��*ង" H �ប"���� : DEF �

E. �គ ! ,,a b c'ច�ន�នព��ម�នF# �ជ%&នប�).5�ផ7GងH7 �" 2 2 24 9 14a b c+ + = �

Page 8: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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បfg ញJ� 3 128b c abc ≤+ + �

K. �គ !Fន�គមនj� 22011

20112011

0

( ) ( 2011 ) (1 )k k k

k

k x CF x xx −

=−= −∑ �

�ក��C5ធ�ប�ផ���ប"Fន�គមនj�2�5/ច�63 � [ ]0;1 � '()&'()&'()&'()&

ចំេល�យ

�. 5កq:r � 2x ≥ −

ម� �មម*5នDង� 2 22 4) 2 4 (2)9 ( 2)( 2 2( 2)x x x xxx− + + +=

−+ +

��យ 2 22 4 ( 1) 3 3x xx − + = − + ≥

)ចកFងRS�ងព��Cនម� � (2)នDង 2 2 4x x− + , �យ/ង;ន�

2 2

2 24 9 2 0

2 4 2 4

x x

x xx x

+ + − + = − + − +

2

2 19 109

2 164x

xx

x

+ = =⇒ ⇒ ±− +

(យក)

�. ម� �).5 !មម*5នDង�

( ) ( ) ( ) ( )2 2 2 22010 2012 3 2011x x y z x+ = +−+ + + −

( ) ( ) ( ) ( )2 2 2 22010 2011 2012 3x x x y z⇔ + + =+ + + + −

2 2 2 2 212066 2010 2011 2012 ( 3)3 x y zx⇔ + + + + = + −

FងR�ង�ឆBង�ប"ម� �)ចកនDង 3;ន�:5"�n/ 2, FងR�ង�- ��ប"ម� �

)ចកនDង 3;ន�:5"�n/ 1 � M�n/ 0�

.*ច�ន� ម� �).5 !An នU'ច�ន�នគ�"�

1. ).a .�ប*ង�យ/ង��យបI% ក" Lemma : " ! ,X Y 'ព��ច�ន�ច�2�5/�ងBង" ( )O , �ងBង" ( ')O

ម�យ ប9�នDង XY ��ង" U �O/យប9�ក��ងនDង ( )O ��ង" V � �ព5�6�, ប67 �" UV �"=ម

ច�ន�ចក,- 5 Z �ប"ធ�* XY ម�ន&នផ7�ក "V �

Page 9: Vnmo 30 4-2011-grade 10

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ព��'.*ច�ន�, ព�ន��!ចtប"ប�)5ង\�ងផa�� ( '): ( )V O O→ � �ព5�6� XY d→

�ផ7GងH7 �" ||d XY �O/យ d ប9�នDង ( )O ��ង" Z '�*បiព�ប" U Z⇒ 'ច�ន�ចក,- 5

�ប"ធ�* XY �

ព�ន��!5�l�").5 !�

=ង 1 1 1, ,A B C 'ច�ន�ច�បពB�ប" , ,DK EM FN �PនDង ( )O , =ម Lemma �យ/ង

;ន 1 1 1, ,A B C 'ប,- ច�ន�ចក,- 5�ប"ធ�* , ,CBB C AA ACB � =ង 0 0 0, ,A B C

'ប,- ច�ន�ចឆ3���ប" 1 1 1, ,A B C �ធ@បនDង O , �ព5�6� 0 0 0A B C∆ , 1 1 1A B C∆ &ន

ប,- �ជmង�បA� �

មu9ង�ទ@�� 0 0 0 0 0 0, ; , ; ,C EF AI A C FB D BI A B DE CI⊥ ⊥ ⊥

Sញ;ន 0 0 0,A B C DEF∆ ∆ &នប,- �ជmង�បA� � .*ច�6� 1 1 1,A B C DEF∆ ∆

&នប,- �ជmង�បA� �O/យម�ន�n/A� ( 1 1 1A B C∆ \�Dកក��ង ( ),O DEF∆ \�Dកក��ង ( )I )

⇒ ∃ចtប"ប�)5ង\�ង ប�)5ង DEF∆ �P' 1 1 1A B C∆

1 1 1, ,EBDA FC⇒ �បពBA� ��ង"ផa�� P �ប"ចtប"ប�)5ង\�ង�

).b ព��ន�� ). , ,a P O I⇒ ��"��ង"ជ��A� (1)

=ង ', ', 'A B C 'ប,- ច�ន�ច�បពB�ប"ប,- ក�ព" ', ', 'DD EE FF �ប" DEF∆

នDង (1)� ��យ ' 'EE FF \�Dកក��ង�6� � � � �B A D B ED C FD C A D D′ ′ ′ ′ ′ ′= = = ⇒ '

ច�ន�ចក,- 5�ប"ធ�* |' |' B C ID B CB C C B′ ′ ′⇒ ⊥ ⇒ (��Z� BC ID⊥ )

.*ចA� ).�� ' ' || , ' ' ||C A CA A B AB , �O/យ H 'ផa���ងBង"\� Dកក��ង A B C′ ′ ′∆

ព��6� ,ABC A B C′ ′ ′∆ ∆ &នប,- �ជmង�បA� ន�ងម�ន�n/A� (��Z� ABC∆ \�Dក

BY

X

'O

O

V

Z B

A

C

1A

1B

1C

0B

0A

0C M

N

K

D

2O E

I

3O

1O

O

Page 10: Vnmo 30 4-2011-grade 10

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ក��ង '( ), BO A C′ ′∆ \�Dកក��ង ( )I ) ⇒ ∃ចtប"ប�)5ង\�ង ប�)5ង ABC∆ �P'

A B C′ ′ ′∆ �

6� ! ', ', 'AA BB CC �បពBA� ��ង"ផa��ប�)5ង\�ង Q Q⇒ , ,I O ��"��ង"ជ��A� (2)

មu9ង�ទ@�, ,I H 'ផa��\� Dកក��ង , , ,ABC A B C Q H I′ ′ ′∆ ∆ ⇒ ��"��ង"ជ��A� (3)

ព� (2)(1), , (3)�យ/ង;នបIg ��e#;ន��យបI% ក"�

E. # �មiព).5 !មម*5នDង� 24 )16 (16 2b c abc ≤+ +

+ Fន�#�-នj# �មiពក*�� �យ/ង;ន�

2 2 2 23( 1) 8( 1)6 116 3 81 b c bb cc ≤ + + + = ++ +

2 2 2 2 2 2 2 2 24 9 )11 ( 25b c a b c a ba c+ + − − − − −+ = −= �6��./ម0���យបI% ក" (1) �យ/ង�Aន")���e# ���យបI% ក"# �មiព�

2 2 2 1 2 (2)b c abca + + − ≥ , ច��Z� , ,a b c ម�នF# �ជ%&ន�

+ =មប�^ប" 2 2 24 9 14a b c+ + = ,# �មiព (2)bច����k/ង# �ញ.*ច�ង�� ម�

( ) ( )2 2 2 2 2 24 914 28a ab c b c abc+ + + + ≥−

2 2 210 51 23 8b c abca⇔ + + ≥

( )2 2 2 2 2 210 5 4 9 2813 14a a abb c cc b⇔ + + + + ≥

+ Fន�#�-នj# �មiពក*��ម-ង�ទ@�ច��Z� 28ច�ន�ន�យ/ង;ន�

( ) ( ) ( )13 20 52 2 2 2 2 2 13 104 512813 10 5 28 28a a b c ab c b c+ =+ ≥

ន�ង ( ) ( ) ( )24 92 2 2 2 2 2 4 914144 9 14 14a a b cb c cab+ =+ ≥

+ .*ច�6�

( ) ( )22 2 2 2 2 2 13 10 5 4 914 1410 5 213 14 28 48 1b c b c b ca a ca a b abc=+ + + + ≥

# �មiព�ក/�&ន�ព5 ( , , ) (1,1,1)a b c = , 5�l�"��e#;ន��យបI% ក"�

K. �យ/ង&ន� 2

0

( ) (1 )n

k k n kn

k

A k nx C x x −

=−= −∑

Page 11: Vnmo 30 4-2011-grade 10

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2 2

0 0 0

(1 ) (1 ) (1 )( ) 2n n n

k k n k k k n k k k n kn n n

k k k

C x x k C x x kC x xnx nx− − −

= = =− + − −= −∑ ∑ ∑

ព�ន��!� 10 1

(1 ) (1 )n n

k k n k k k n kn n

k k

kC xA x kC x x− −

= == − = −∑ ∑

1 11 1

1 1

(1 ) (1 )n n

k k n k k n kn n

k k

C x x nx C xn − − − −− −

= =− = −= ∑ ∑

[ ](1 ) n knx x x nx

−= + − =

2 22

0 1

(1 ) (1 )n n

k k n k k k n kn n

k k

k C x x k C x xA − −

= == − = −∑ ∑

11

1

1 11 1

1 1

11

1

2 22

2

(1 )

(1 ) ( 1) (1 )

( 1) (1 )

( 1) (1 ) ( 1)

nk k n kn

k

n nk k n k k k n kn n

k k

nk k n kn

k

nk k n kn

k

C x x

C x x n k C x x

k C x x

nx n

n

n

n C x x nx n n

x

x

n n

− −−

=

− − − −− −

= =

− −−

=

− −−

=

=

=

= +

− + − −

− −

= + − − = + −

∑ ∑

[ ]30

() 1 )(1 1n

nk k n kn

k

C x xA x x−

=+ −= − = =∑

.*ច�ន� 2 2 2( 1) 2( ) (1 )( ) nx n n x nA n nxx x x+ + − − = −=

Fន�#�-នj5ទQផ5�ង�5/�យ/ង;ន� ( ) 2011 (1 )f x x x= −

��យ [ ];10x∈ �6� 0,1x x− ≥ � ព��6� =ម# �មiពក*���

2

2011.(1 ) 2011

( )2 4

x xf x ≤ =+ −

, Iw មiព�ក/�&ន�ព5 1

2x =

.*ច�ន� 2011max ( )

4f x = ទទ�5;ន�ព5 1

2x = �

'()&'()&'()&'()&

Page 12: Vnmo 30 4-2011-grade 10

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Page 11

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២

�. �����យ�បព<នQម� �� 21 1 4( ) 3.

30 4 2011

x y x y x y

x y

+ + + = + +−

+

=

�. �ក�គប"ប,- ច�ន�ន).5&ន�5ខប�ខ7ង")ចក�ច"នDង 1189ង, !ផ5)ចក�ប"?

�n/នDងផ5ប*ក ���ប"ប,- �5ខ=មខ7ង"�ប"?�

1. �គ !���� : ABC ).5 A'ម���*ចប�ផ��, \� Dកក��ង�ងBង" ( ),O ច�ន�ច D ច5<��2�5/

ធ�*�*ច BC � �ម.uទ<��ប" AB ន�ង AC �" AD ��@ងA� ��ង" E ន�ង F � =ង T '

ច�ន�ច�បពB�ប" BE ន�ង CF � ប67 �" �"=ម T �O/យ�បនDង AB �" AD ��ង"

N � ង"�ប�5k*� ម TNDM � =ង P 'ច�ន�ចក,- 5�ប" MC �O/យ I 'ច�ន�ច

�បពB�ប" PT នDង�ម.uទ<��ប" OP �

��យបI% ក"J I ច5<��2�5/)ខpនDងម�យ�

E. �គ ! , ,x y z 'ប,- ច�ន�នព��# �ជ%&ន�ផ7GងH7 �"ប,- 5កq:r � 2 11

x y+ ≤ ន�ង 4

2yz

+ ≤ ,

�ក��C5�*ចប�ផ���ប"ក�នyម� ( , , ) 9P x y z x y z= + + �

K. ��យបI% ក"J� ក��ង 17ច�ន�នគ�",កL��យ )�ង&ន 9ច�ន�ន).5&នផ5ប*ក

)ចក�ច"នDង 9�

N. ប,- ក�ព*5�ប" ABC∆ 'ច�ន�ច).5&នក*F����ន'ច�ន�នគ�" �O/យម�ន&ន

���� :,�*ច'ង ABC∆ ន�ង&ន^ង.*ចនDង ABC∆ �ទ ច��Z�ក�ព*5�ប"'

ប,- ច�ន�ច&នក*F����ន'ច�ន�នគ�"� បfg ញJ ផa�� D �ប"�ងBង"\� Dក��]�ប"

ABC∆ ម�ន)មន'ច�ន�ចម�យ).5&នក*F����ន'ច�ន�នគ�"�ទ �

'()&'()&'()&'()&

Page 13: Vnmo 30 4-2011-grade 10

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Page 12

ចំេល�យ

�. �����យ�បព<នQម� �� 2

(1)30 4 201

1 1 4( ) 3

1

.x y x y x y

x y

+ + + = + +

+− =

=ង 0u x y= + ≥

2 4

2

2

1 (2)

30

1

34 20

4 3

11) 1( u y

x y

u u

u

u + +

⇔ −

+ = =

+ =

2 4 2 4(2) 11 04 113 3 4u u u u u y+ =⇔ + + ⇔ −+ − =+

2 22 2

2

2 2

2

2

1 3)(1 2 ) 0

1

)1

0 1 2(

(1 23

1(1 2 1 2 0

) 0

3

1 2

uu

u u

u uu

uu

x y

u

u

+ −⇔ + =+

⇔+

⇔ = ⇔

+ −+

− + + =

+

+

− =

Sញ;ន� 2 2 1 2013 998;

30 4 2011 34 17

x y

x yx y

+ = ⇔ = = −

− =

.*ច�ន� U�ប"�បព<នQគM 2013 998;

34 17x y = −= �

�. =ង x abc= 'ច�ន�ន).5��e#�ក�

�យ/ង;ន�

2 2 2) (1)

0 ; ;

10

9

0

;

0 11(

;

0 1 b c

a b c

a

a b

a c a

c

b + + = + +≤

�យ/ង&ន� 2 2 2(1) 99 11 )( ) 11(a b a c b b ca+ + + − + +=⇔

11a c b k⇔ + − = )� 18 8 0a c b k− ≤ + − ≤ ⇒ = � M 1k =

+ 0 0k a c b b a c⇒ ⇒= + == +−

Page 14: Vnmo 30 4-2011-grade 10

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Page 13

2 2 2 2 2 2

2 2

( )(1) 9 9

10 2 ) 2( ( )

b c aa b a c c

ac c c n n

a a c a

a c a

⇔ + + ⇔ + + +

+ = + + =

+ + =+ ⇔ ∈= ℕ

2 2 2 22 4 )10 2 2 (2 5( ) 4 0a na n a a n nn na⇔ + + ⇔ + − + −= =+

5កq:r � 20 16 25 012n n∆ ≥ ⇒ − + ≥−

4 91 4 91

60

6n n

− − − +⇒ ≤ ≤ ⇒ = 0;bc a⇒ = = ន�ង 2 5 0a a− =

5a b⇒ = = ន�ង 5 00 5xc ⇒ ==

+ 1k = , �យ/ង;ន 1 11 1b a ca c b ⇒ = += −+ − �6�

2 2 2(1) 99 11 11 11 )(b ba ca + + ++⇔ =

2 2 2 2 2 2

2 2

( 11)

2 121 2 22 2

9 1 9 11 1

10 10 2 2

b c a ca b a a a c c

c ac a c

a

a c a

⇔ + + ⇔ + + − +

⇔ + + + −

+ + = + + − + =

−+ − =

2 22 2 32 22 3 131 0c ac a ca c⇔ + + − + + = ⇒ 'ច�ន�ន�

2 1c n⇔ = + .

.*ច�6��យ/ង;ន� 2 2(2 15) 4 19 55 0n a n na + − + − + =

�យ/ង&ន� 212 16 25 0n n∆ = − + + ≥

{ }4 91 4 910;

6 61; 2n n⇒ ≤ ≤ ⇒ ∈− +

2 5; 6bc an• = ==⇒ − ន�ង 2 11 33 0aa − + = (An នU)

1 3; 8bc an• = ==⇒ − ន�ង 2 13 40 0aa − + =

8; 0ba⇒ = = ន�ង 8 33 0xc ⇒ ==

1; 10 0 0c bn a⇒ = = −• <=

.*ច�ន� ប,- ច�ន�ន).5��e#�កគM 550x = ន�ង 803x = �

1. + =ង H 'ច�ន�ច�បពBទ�ព���ប" BE នDង�ងBង" ( )ABC �

K 'ច�ន�ច�បពBទ�ព���ប" CF នDង�ងBង" ( )ABC �

�យ/ង;ន� AD CK= ន�ង AD BH=

BKHC⇒ 'ច��� :Z� យម;��

N

B

A

C

D P

M

K E O

F

I

T

Page 15: Vnmo 30 4-2011-grade 10

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Page 14

TBTB TT C TK TC CK ADK + = + =⇒ == ⇒ .

+ �យ/ង&ន TB NA ND TC⇒= = )� ND TM TC TM⇒= =

TMC⇒ ∆ ម;���ង" T

�� �

�0 0180 180

2 2

MTC AFTTCM OFC= − =−

⇒ =

||CM OF⇒ .

M⇒ 4���2�5/ប67 �" ( )∆ �"=ម C �O/យ)កងនDង AC �2នDង�

P⇒ ច5<��2�5/ ( )∆ �2នDង �

+ ��យ I 4���2�5/�ម.uទ<��ប" OP �6� IO IP= )� ( ; )IP d I= ∆ (��Z� IP

)កងនDង ( )∆ )

Sញ;ន I )�ង4���2�5/;9 9̂ប*5).5&នក�ន��'ច�ន�ច O �O/យប67 �"�;ប"ទ�

គM ( )∆ , �ព5 D ច5<��2�5/ធ�*�*ច BC �

E. ��យ , ,x y z # �ជ%&ន�6��យ/ង;ន� 2 1;2)1 (1

11y y

x y y≥ ⇒ ⇒ ∈+ > >

�យ/ង&ន� 1

2

2 1 1 2 22

1 1

4 4

2

xy

y z

y

x y y y y

z y

−= =≤ − ⇒ ≥−

− ≥

+

−≤

.*ច�6� 2 4( , , ) 9 2

1 2P x y z y

y y+ + +

− −≥

21 4

2 9( 1) 9(2 ) 2 61 2

y yy y

= + − + + − + −

≥−

Iw " "= �ក/�&ន 2 4 1; ; 9( 1)

1 2 1

yy

y y yx z = −

− − −⇔ = =

ន�ង 4 49(2 ) 8,

2 3y yx

y= − =⇔ =

−ន�ង 6z =

K. + �យ/ង��យបI% ក" Lemma �ង�� ម�

" ក��ង 5ច�ន�នគ�",កL��យ, )�ង&នប�ច�ន�ន).5&នផ5ប*ក)ចក�ច"នDង 3" �

�^យ�

Page 16: Vnmo 30 4-2011-grade 10

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Page 15

ក�:� ទ�ម�យ� ក��ង 5ច�ន�នគ�"�ង�5/, &នប�ច�ន�ន�k/ង ).5&ន�:5"��មA� �ព5

)ចកនDង 3�6� Lemma �ង�5/គMព���

ក�:� ទ�ព��� ក��ង 5ច�ន�នគ�"�ង�5/, &ន��ច'ងប�ច�ន�ន).5&ន�:5"��មA� �ព5

)ចកនDង 3�6�នDង&នប�ច�ន�ន).5&ន�:5"��@ងA� គM 0;1; 2�ព5)ចកនDង 3�

�ព5�6� ផ5ប*កCនប�ច�ន�ន�ន�)ចក�ច"នDង 3� Sញ;ន Lemma ព���

.*ច�ន� Lemma ��e#;ន��យបI% ក"�

+ Fន�#�-នj Lemma �ង�5/�យ/ង;ន�

• យក 5ក��ង 17ច�ន�នគ�").5 ! �6�ក��ង 5ច�ន�ន).5��e#;នយក &នប�ច�ន�ន).5

&នផ5ប*ក)ចក�ច"នDង 3� =ង �បព<នQប�ច�ន�ន�ន���យ 1 1 1; )( ;a b c ន�ង=ង

1 1 1 1a cm b= + + �

• យក 5ក��ង 14ច�ន�នគ�"�25" �6�ក��ង 5ច�ន�ន).5��e#;នយក &នប�ច�ន�ន).5

&នផ5ប*ក)ចក�ច"នDង 3� =ង �បព<នQប�ច�ន�ន�ន���យ 2 2 2; )( ;a b c ន�ង=ង

2 2 2 2a cm b= + + �

• យក 5ក��ង 11ច�ន�នគ�"�25" �6�ក��ង 5ច�ន�ន).5��e#;នយក�ន� &នប�ច�ន�ន

).5&នផ5ប*ក)ចក�ច"នDង 3� =ង �បព<នQប�ច�ន�ន�ន���យ 3 3 3; )( ;a b c ន�ង

=ង 3 3 3 3a cm b= + + �

• យក 5ក��ង 8ច�ន�នគ�"�25" �6�ក��ង 5ច�ន�ន).5��e#;នយក�ន� &នប�ច�ន�ន

).5&នផ5ប*ក)ចក�ច"នDង 3� =ង�បព<នQប�ច�ន�ន�ន���យ 4 4 4; )( ;a b c ន�ង=ង

4 4 4 4a cm b= + + �

• ក��ង 5ច�ន�នគ�"�25", &នប�ច�ន�ន).5&នផ5ប*ក)ចក�ច"នDង 3� =ង�បព<នQ

ប�ច�ន�ន�ន���យ 5 5 5; )( ;a b c ន�ង=ង 5 5 5 5a cm b= + + �

ក��ង 5ច�ន�នគ�" 1 2 3 4 5; ; ; ;m mm m m &នប�ច�ន�ន).5&នផ5ប*ក)ចក�ច"នDង 3�

zប&J ប�ច�ន�ន�6�គM ; ;i j km m m � �ព5�6� 9ច�ន�ន 1; ; ; ; ; ; ; ;i i j j j k k kb ca a b c a b c

&នផ5ប*ក)ចក�ច"នDង 9 (បIg ��e#;ន��យបI% ក")�

Page 17: Vnmo 30 4-2011-grade 10

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N. �យ/ងbច��ជ/�� /�បព<នQក*F����ន Oxy 89ង, ! , ,A B C &នក*F����ន

( ;( ), (0 , ;; )0 ) B a b C c dA ).5 ; ; ;a b c d 'ប,- ច�ន�នគ�"�

zប&J ( , )D x y 'ផa���ប"�ងBង" ( ),ABC D 'ច�ន�ច&នក*F����ន'ច�ន�នគ�"�

�យ/ង;ន� 2 2 2 2 2 2( ) ( )BD yAD x x a y b= ⇒ + = − + −

2 2 2 2 0a b ax by⇒ + − − =

.*ច�6� 2 2a b+ 'ច�ន�នគ* ,a b⇒ &ន5កq:�គ*�.*ចA� a b⇒ + ន�ង a b−

'ច�ន�នគ* �

.*ចA� ).�, ,c dd c+ − 'ច�ន�នគ*�

2 2( ) )

2

(

4

a bb a b a− −=⇒+ 'ច�ន�នគ�"�

.*ច�ន� ប,- ច�ន�ច ;2 2

a b a bX

+ − =

ន�ង ;2 2

c d c dY

+ − =

'ប,- ច�ន�ច&ន

ក*F����ន'ច�ន�នគ�"�

)� 2 2 2 2 2

2 ( ) ( )

4 4 2 2

a b a b a ABAX

b+ −+ =+= = , .*ចA� ).��យ/ង&ន� 2

2

2

ACAY =

2 2

2

2 2

a b c d a b c dXY

+ − − − − + +

=

( ) ( )2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2

1( )

( ) 2( )( ) ( ) ( ) 2( )( )

2 2 2

( ) ( ) ( )41

( )41

241

2 2 2 2 2 2 2 2

2 2 224

( )

2

4 4

( )

2

a d b c a d a d b c a d b c

d b c ab ac

b c a d a d b c

b

bd cd a ac bd cd

b c d ac bd

b d

c

a

a

a c BC

= − + − + − +

= −

=

=

+

+ − + − − + + + + − + +

+ + + + − − + − − −

−= =

+ + + − −

+ −

6� ! AXY∆ &ន^ង.*ចនDង ABC∆ ន�ង AXY �*ច'ង ABC∆ , ផ7�យព�ប�^ប"�

.*ច�ន� D ម�នbច'ច�ន�ច).5&នក*F����ន'ច�ន�នគ�"�ទ�

'()&

Page 18: Vnmo 30 4-2011-grade 10

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Page 17

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទ៣ី

�. �����យ�បព<នQម� ��ង�� ម� 2 2

2 2

4 4 12 11 0

4 2 4 12 0

y x y

x y xy x y

x + − + + =

+ − − + − =

�. �កU'ច�ន�នគ�"�ប"ម� �� 4 4 41 2 15... 1215x xx + + + = �

1. ���� : ABC �ផ7GងH7 �"5កq:r FB��ប/?�ផ7GងH7 �"5កq:r �ង�� ម� cos cos cos 17

8 15 17 120

A B C+ + =

E. �គ ! , ,a b c 'ប�ច�ន�នព��# �ជ%&ន�ផ7GងH7 �" 1abc = � បfg ញJ�

1 1 1 1 1 1 9

1 1 1 2a b c a b c + + + + + + +

K. ក��ង�ន�� { }...1 ,,2 10, 20T = ��/&នប9�6n នច�ន�ន).5)ចកម�ន�ច"នDង 2,3,5,7,11 ?

N. �គ !ច��� :)កង ABCD &ន M 'ច�ន�ចក,- 5�ប" AB , N 4���2�5/កន3�

ប67 �"ព���ប" BCD� =ង P 'ច��,5)កង�ប" N �2�5/ BC �

��យបI% ក"J �ប/ MN DP⊥ �6����� : ANDម;�� '()&'()&'()&'()&

ចំេល�យ

�. �បព<នQ).5 !មម*5នDង�

2 2

2 2 4 4 12 11 0 (1)4 4 12 11 0

3 23(2 8) 3 23 (2)

2 8

x y x yy x y

xx y x y

x

x+ − + + =+ − + + = ⇔ −+ =

− =

+

ជ�ន� (2)ច*5 (1) �យ/ង;ន�

2

2 3 23 3 2314 1 0

2 8 8.

24 12

x xx x

x x+ −− − + = + +

+

4 3 2

2

4

1( 1)( 4)(4

16 88 720 620 0

32 204) 04

xx

xx x

x x

xxx

⇔ + + − + =

⇔ + +=

− − =⇔

=

Page 19: Vnmo 30 4-2011-grade 10

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+ ច��Z� 1 2x y= ⇒ = −

+ ច��Z� 3 1x y= ⇒ = −

.*ច�ន� ម� �&នច��5/យព�� (1; 2),(3; 1)− − �

�. �កU'ច�ន�នគ�"�ប"ម� �� 4 4 41 2 15... 1215x xx + + + =

+ ច��Z� 4 42 : 16 0 (mod16)i ik x kx = = ≡

+ ច��Z� 2 1ix k= + , �យ/ង;ន� 4 2 21 ( 1)( 1)( 1) 4 ( 1)( 1)i i i i ix x x k xx k− = − + + = + +

��យ 22;( 1) 1 2ik k x ++ ⋮ ⋮ (��Z� ix �) �6� 4 1 16ix − ⋮ � M 4 (mo1 d16)ix ≡

�យ/ង;ន� 15

4

1

(mod16)ii

x r=

≡∑ ក��ង�6� 0 15r≤ ≤ �

មu9ង�ទ@� 1215 75.16 1 15 (mo 165 d )= + ≡ , .*ច�6� ix �ច��Z��គប" 1,15i = �

�O/យ��យ 4 2401 17 12 5= > �6� 5ix i≤ ∀ �

��យ 4 45 12505 1215+ = > �6�&ន��ច/នប�ផ��ច�ន�ន ix ម�យ, zប&JគM 15x ,

�ផ7GងH7 �" 4 415 5 625x = = �

�ព5�6� 4 4 41 2 14... 1215 625 590x xx + + + = − = ន�ង 3ix ≤ 1,14i∀ = ,

)� 4.59 3 20 7 3= + �6���e#&ន��ច/នប�ផ�� 7ច�ន�នក��ង 14ច�ន�ន 1 2 14, , ...,xx x �ផ7Gង

H7 �" | | 3ix = , )� 4590 83 .< , ម�នម�O��ផ5

.*ច�ន� | | 3ix ≤ ច��Z��គប" 1,15i = �

�យ/ង;ន� 4 4 4 41 2 15..1215 . 15.3 1215x x x+ + + ≤ == , .*ច�6� | | 3ix = �គប" 1,15i = �

.*ច�ន� ម� �).5 !&ន 152 U'ច�ន�នគ�"&ន^ង ( )3; 3; ... ; 3± ± ± �

1. cos cos cos 15.17cos 8.17cos 8.15cos

8 15 17 2040

A B C A B C+ ++ + =

2 2 28 17

4080

1

20

17

1

5 =+ +≤

ព��'.*ច�ន� 2 2 22.8.(17cos 15cos ) 15 17 2.15.17c s8 oB C A− + + + −

[ ]2 2 2 215 17 2.15.17co8 (17cos 15cos ) s (17 cos 15cos )B C A B C= + − −− + + +

Page 20: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 19

)� 2 2 217 2.15.17cos (17cos 15cos )15 A B C+ − − +

2 2 2 2

2 2 2 2

2

15 sin sin

sin

17 2.15.17(cos cos cos )

15 17 2.15.17sin sin

(15sin 17sin )

in

0

s

C B A B C

C B C B

C B

+= − +

= + −

= − ≥

.*ច�ន� cos cos cos 17

8 15 17 120

A B C+ + ≤

មiព�ក/�&ន5���=)�� 15sin 17sin

8 17cos 15cos

C C

B C

= = +

15 1715 17 8sin sin

15sin sin sin sin8 cos 15cos

sin

B CC B C A

B CB

= = = = +

⇔ ⇔

� M 15 17 8

b c a= = �

.*ច�6� ���� : ABC &ន^ង.*ចនDង���� :).5&ន�ជmងS�ងប��n/ 8,15,17 �

��យ 2 2 215 78 1+ = , 6� ! ���� : ABC )កង��ង" C �

E. 1 2 31 1 1 1 1 1

1 1 1T T

a b c a bT T

c = + + + + = + + +

+

+

).5 11 1 1

(1 ) (1 ) (1 )T

a a b b c c+ +=

+ + +

21 1 1

(1 ) (1 ) (1 )T

b a c b a c+ +=

+ + + ន�ង 3

1 1 1

(1 ) (1 ) (1 )T

c a a b b c+ +=

+ + +

��យ��6ទ� , ,a b c .*ចA� �6��យ/ងbចzប&J a b c≤ ≤ , �ព5�6�

1 1 1

a b c≥ ≥ ន�ង 1 1 1

1 1 1a b c+ +≥

+≥ �

�យ/ង;ន� 1 2T T≥ � ព��'.*ច�ន�

1 21 1 1 1 1 1

(1 ) (1 ) (1 ) (1 ) (1 ) (1 )T

a a b b c c b a c a cT

b

+ + − + + + + + + + +

=

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1a c a b a b c b c = − + − + − + + + + + +

Page 21: Vnmo 30 4-2011-grade 10

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Page 20

1 1 1 1 1 1 1 1

1 1 1 10

a c a b b c b c = − − + − − + + + +

�យ/ង;ន 23

2T ≥ � ព��'.*ច�ន�

��Z� 1abc = �6�=ង , ,y z x

bax y z

c= = = � �ព5�6�

21 1 1

( ) 3x y z

T x y zy z z x x y y z z x x y

+ + = + + + + − + + + + + +

= 3

2≥

.*ចA� ).� 33

2T ≥

.*ច�ន� 9

2T ≥ , មiព�ក/�&ន5���=)� 1a b c= = = �

K. =ង { } { } { }1 2 3/ 2 / 3 / 5; ; ;T k TA k A k k kkA T= ∈ = ∈ = ∈⋮ ⋮ ⋮

{ } { }4 5/ 7 / 11; ;T k A TA k kk= ∈ = ∈⋮ ⋮

�ព5�6� 1 2 3 4 5A A A A A∪ ∪ ∪ ∪ '�ន��ប,- ច�ន�ន4���2ក��ង T )ចក�ច"នDងច�ន�ន

ម�យក��ងប,- ច�ន�ន 2,3,5,7,11�

�យ/ង;ន� 1 2 32010 2010 2010

| 1005; 670;| | | 22 3

| 40|5

A A A= = == ==

4 5 1 2| | |2010 2010 2010

| 287; 182; 335;7 11 6

A A AA = = = =

= = ∩

1 3 1 4 1 52010 2010 2010

201; 143; 91;10 14 22

A A A A A A = = = = = = ∩ ∩ ∩

2 3 2 4

2010 2010134; 95;

15 21A A A A = = = = ∩ ∩

2 5 3 4

2010 201060; 57;

33 35A A A A

= = = = ∩ ∩

3 5 4 5

2010 201036; 57;

33 35A A A A

= = = = ∩ ∩

1 2 3 1 2 4

2010 201067; 47;

30 42A A A A A A = = = = ∩ ∩ ∩ ∩

1 2 5 1 3 4

2010 201030; 28;

66 70A A A A A A

= = = = ∩ ∩ ∩ ∩

Page 22: Vnmo 30 4-2011-grade 10

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Page 21

1 3 5 1 4 5

2010 201018; 13;

110 154A A A A A A = = = = ∩ ∩ ∩ ∩

2 3 4 2 3 5

2010 201019; 12;

105 165A A A A A A

= = = = ∩ ∩ ∩ ∩

2 4 5 3 4 5

2010 20108; 5;

231 385A A A A A A = = = = ∩ ∩ ∩ ∩

1 2 3 4 1 2 3 5

2010 20109; 6;

210 330A A A A A A A A

= = = = ∩ ∩ ∩ ∩ ∩ ∩

1 2 4 5 1 4 5

2 3 4 5 1 2 5

3

3 4

2010 20104; 2;

462 770

2010 20101; 0

1155 2310

A A A A A A A A

A A A A A A A A A

= = = =

= = = =

∩ ∩ ∩ ∩ ∩ ∩

∩ ∩ ∩ ∩ ∩ ∩ ∩

)���យ 1 25

41

5

31

5 i ki

ij

jki

A A A A A A A A A≤ < < ≤=

= − −∑ ∑∪ ∪ ∪ ∪ ∩ ∩

1 2 4 51

35

ii j

j k qk q

A A A A A A A A A≤ < < < ≤

− +∑ ∩ ∩ ∩ ∩ ∩ ∩ ∩ 1593=

.*ច�ន� �ន�� T &ន 2010 1593 417− = ច�ន�ន).5ម�ន)ចក�ច"នDង 2,3,5,7,11 �

N. ��ជ/�� /����យ89ង, ! ; ,B O OxA yO D≡ ∈ ∈

�ព5�6� (0; ), ( ;0), ( , )(0;0),B b D d C d bA , ( ; ), ( ;,2

)0; N t t b d P t bb

M

+ −

. 0DP MN DPMN ⊥ ⇔ =� �

2

2( ) 0 02

(2

)b db b

t t d b t d t bdt − + + − = − = ⇔ ⇔ −

+ +

2 22 2 2( ) ,t t b Dd dAN A= + + − =

�យ/ង;ន� 2 2 2 2 2( )AN AD AN dAD t t b d⇔ = ⇔ = + + −=

2

2 ) 02

(b d tb

t bd−⇔ + − =+ (បIg ��e#;ន��យបI% ក") �

'()&'()&'()&'()&

Page 23: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 22

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី៤

�. �កU'ច�ន�នព�� ;;x y z �ប"�បព<នQម� �� (1)

9(2

1

)4

x y z

x yz y z x

x y

x y

z

z+ + =

+ + =

+ + +

�. �ក�គប"ច�ន�នគ�"# �ជ%&ន , ,p q n ច��Z� ,p q 'ច�ន�នប[ម �./ម0� !�

( 3) ( 3) ( 3)p p q q n n+ + + = +

1. ក��ងប3ង"�គ !ច�ន�ច P �2នDង� ព�ន��!���� :)កងម;� ABC (&ន � 090 ;ABC =

5AB < ) �ផ7GងH7 �"5កq:r 2PA = ន�ង 3PB = �

�ក��C5ធ�ប�ផ���ប"Fង̀�" PC �

E. �គ !ប,- ច�ន�នព�� ;;a b c �ផ7GងH7 �"�

; 9; 36 ;12 360 48b c c bc c bcc ca a≤ ≤ ≥ ≥ + ≥ + +< �

��យបI% ក"J 0a b c+ − ≤ , មiព�ក/�&ន�2�ព5,?

K. ក��ង�ន��ប,- ច�ន�នគ�"ធមn'��ព� 2.5" 2011, �គ��ជ/យក 1006ច�ន�ន �O/យប�ង̀/�

'�ន���ង { }1 2 1006; ; ... ;A aaa= �

��យបI% ក"J ក��ង A&នយក;ន 2ច�ន�ន �./ម0� !ច�ន�នម�យ�ន�)ចក�ច"នDងច�ន�ន

ម�យ�ទ@� �

N. �គ ! 6ច�ន�នព�� ; ; ;; ;b c d ea f )�ប�ប|5 �ផ7GងH7 �"5កq:r � 2 2 2 2 2 22 2 3; 10 6 33 0b a c d c f e fa e+ − = + − = + − + + =

2 2 2 2 2 2 12;c b d ac da b+ + + − − = ន�ង ( 5)( 2 ) ( 3)( 2 ) 0e a c e f b d f− + − + + + − = �

ក�:�"��C5�*ចប�ផ���ប"ក�នyម 2 2

2 2

a b dS e f

c ++ + = − −

'()&'()&'()&'()&

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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ចំេល�យ

�. ��យ ; ;x y z # �ជ%&ន�6�

1 1 (*)xy xz yz yx zx zy

x y zz y x z y x

+ = + + =⇔+

=ង tan ; tan ; tan2 2 2 2

A yz B zx C xy

x y= = = ).5 **); (0 ;A B C π< <

ម� � (1)bច����k/ង# �ញ'� tan tan tan tan tan tan 12 2 2 2 2 2

(3)A B B C C A+ + =

�O/យម� � (2)bច���'� 2 2 2

1 1 1 9

41 tan 1 tan 1 tan2

4)

2

(

2A B C

+ + =+ + +

(3) tan tan tan 1 tan tan tan cot2 2 2 2 2 2 2 2

A B C B C A B C + = − = +

2kA B C+ + =⇔ π + π , =ម (**) �6� A B C+ + = π

�ព5�6�

2 2 2 9 3 cos cos cos 9(4) cos cos cos

2 2 2 4 2 4

A B C A B C+ + ++ + = =⇔ ⇔

2 31 2s

3co in 2sin cos cos cos s

2 22 2 2

A BB C

A CA + + = −⇔ ⇔ − + =

24sin 4sin cos 1 02 2 2

2sin cos2 2

3sin 0

2

A A B C

A B C

BA B C

C

−− + =

− = − =

π⇔ ⇔ = = =

�បព<នQម� �&នច��5/យ 1

3x y z= = = �

�. ច��Z� m'ច�ន�នគ�"# �ជ%&ន, �យ/ង�ក;ន�:5"�ព5)ចក ( 3)m m + នDង 3

ព�ន��! 3ក�:� �

3m k= �6� 0( 3) 3od )(mm m ≡+

Page 25: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 24

3 1m k= + �6� 1( 3) 3od )(mm m ≡+

3 2m k= + �6� 1( 3) (mod 3)m m ≡+

.*ច�ន� �ព5 m )ចក�ច"នDង 3�6� ( 3)m m + )ចក�ច"នDង 3, �ព5 m )ចកម�ន�ច"

នDង 3�6� ( 3)m m + )ចកនDង 3;ន�:5"�n/ 1

��kប"មក5�l�"�យ/ង# �ញ, �យ/ង&ន ( 3) ( 3) (mod( 3) 3)n np p q q ≡ ++ + +

+ ក�:� �: �ប/ p ន�ង q �ទQ)�)ចកម�ន�ច"នDង 3�6�

( 3) ( 2 (mod 3)3)p p q q+ + + ≡ .

ក��ង�ព5�6� 0( 3) 3od )(mn n ≡+ � M 1( 3) 3od )(mn n ≡+

.*ច�ន� ក�:� �ន�ម�ន�ផ7GងH7 �"�

+ ក�:� �: ក��ងព��ច�ន�ន ,p q &នម�យ)ចក�ច"នDង 3

zប&J p )ចក�ច"នDង 3, )� p 'ច�ន�នប[ម�6� 3p =

�ព5�6� 18 ( 3) ( 3)q q n n+ + = + , Sញ;ន 18 ( 3) ( )3 (*)n n q q= + − +

��យ n q> �6� ( 1)( ( 4) (*3 *) )n n q q q ++ ≥ +

ព� (*) & (**) �យ/ង;ន 18 2 4 7qq≥ + ⇒ ≤

.*ច�ន� {2; 3; 5; 7}q∈ (��Z� q 'ច�ន�នប[ម)

ច��Z� 2q = �6� 18 10 3) 4(n n n= + ⇒ =+

ច��Z� 3q = �6� 18 18 ( 3)n n+ = + ⇒ម�ន&ន n

ច��Z� 5q = �6� 18 40 ( 3)n n+ = + ⇒ម�ន&ន n

ច��Z� 7q = �6� 18 70 3) 8(n n n= + ⇒ =+

ព�ន��!.*ចA� ).� �ព5 3q =

.*ច�ន� �យ/ង;នប,- UគM� 3 3 2 7

2 ; 7 ; 3 ; 3

4 8 4 8

p p p p

q q q q

n n n n

= = = = = = = = = = = =

1. �យ/ង&ន PC )#ងប�ផ���ព5 P ន�ង C �2&q ង&� ក"�ធ@បនDងប67 �" AB �

ង"���� :)កងម;� APQ ( ;AP QPQ= ន�ង B 4���2�ង)�ម�យ�ធ@បនDង AP ) ,

Page 26: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 25

��ចព�ន��!ចtប"បងB�5 045

AQ (ផa�� ;A ម���ងB�5 045 ) : 'P P֏ ន�ង 'B B֏

ព�ន��!ចtប"ប�)5ង\�ង 2AV (ផa�� A , ផ5�ធ@ប 2k = ) : 'P Q֏ ន�ង 'B C֏

�យ/ង;ន 2PQ AP= = ន�ង 2 ' 'QC P B= 2 3 2PB= =

��យ 2 3 2PQ QC CP PC≤ + ⇒ ≤ +

.*ច�ន� max 2 3 2PC = + ទទ�5;ន�ព5 P 4���2�5/ប67 �" PQ

2 2 24 (2 3 2) 26 12 2AC PCAP = + + = +⇔ = + �

E. =មប�^ប"Sញ;ន�

1

92 (*)

49

4

9

3

b

cb

ac

c

+ ≤

+ + ≤

ប�)5ង� ( )9 9 99 2

4 4

b ba b a c

c c c

+ + = + + + + + −

Fន�#�-នj# �មiព Bunhiakovski �យ/ង;ន�

9 92

4 4 3

b b

c + +

ន�ង 9 93

4 4 3

b ba a

c + + ≤ + +

��ប/�;" (*) Sញ;ន� 3 29 2 3a b c c≤ + ++ + − = +

0a b c⇒ + − ≤ ព��, មiព�ក/�&ន�ព5 4 91; ;b ca = == �

K. + ប�)5ង 1 1 1 2 2 2 1006 1006; ; ... ; im b a m ba a m b= = = , ក��ង�6� ;i im b 'ច�ន�នគ�"# �ជ%&ន,

ib 'ច�ន�ន�&ន��C5ព� 2�P.5" 2011, (ច��Z� 1; ...; 062; 10i = )

+ + + + ព� 2�P.5" 2011&ន 1005ច�ន�ន�).5��e#��ជ/យក 1006ច�ន�ន� ib

+ + + + =ម�ទD-�បទ Dirichlet �6�ក��ងប,- ច�ន�ន ib ).5�ទ/ប��ជ/យក&នព��ច�ន�នគM

j kb b= 2 ; (*)j j j k k km b a m ba⇒ = =

+ + + + zប&J (**)j ka a>

+ + + + ព� (*) ន�ង (**) �យ/ង;ន ja )ចក�ច"នDង ka

Page 27: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 26

N. ក��ងប3ង"ក�:�"��យ�បព<នQក*F����ន�. � Oxy , ព�ន��!�ងBង" 1( )C ផa�� 1(1;0)I ន�ង

� 1 2R = , �O/យ�ងBង" 2( )C ផa�� 2(5; 3)I − � 2 1R =

ព�ប�^ប"�យ/ង;ន ( ; ), ( ; )A a Bb c d 4���2�5/ 1( )C ន�ង ( ; )D e f 4���2�5/ 2( )C

�O/យ 2 3AB = �

�d E 'ច�ន�ចក,- 5�ប"Fង̀�" AB �យ/ង;ន� ;2 2

a c b dE

+ +

ព�ប�^ប"�យ/ង;ន 2 .2 0I D DE =� �

, �ព5�6� 2 2 2 22 2 2 1ED EI R ES I= = − = −

=ង 3( )C '�ងBង"ផa�� 1(1;0)I � 2

23 1 1 4 3 1

2

ABI RR E

= = − = − =

�យ/ង;ន E ��"�2�5/�ងBង" 3( )C

=ង ;H K 'ប,- ច�ន�ច�បពB�ប"ប67 �" 1 2I I នDង 3( )C ( 2 2HI KI> )

�យ/ង;ន 2 2 22 1 2 31 ( ) 1 (5 1) 1 15K IS I I R≥ − = − − = − − =

min 15S = ទទ�5;ន�ព5 E K≡ �O/យ D 'ច�ន�ច�បពB�ប" 2( )C នDង�ងBង" ( ')C

&នFង̀�"ផa�� 2KI �

��យម� �ប67 �" 1 2I I គM 3 4 3 0x y+ − = �O/យ�ងBង" 2 23) : ( 1 1( )xC y− + =

9 3 9 3; ;

5 5 2 5 2 5

9 3 3 3 4 3 9 3 3 3 4 3; ; ;

5 5 5 5

9 3 3 3 4 3 9 3 3 3 4 3; ; ;

5 5 5 5

b c d

d a b

a c b dK

a

c

+ + − = = −

+ − + − − −= + − + − −

⇒ ⇒

= = =⇒

= −=

= =

��យម� ��ងBង" 2 22( ) : ( 35) 1( )yC x +− + = ន�ង�ងBង"

2 2

417 9

( ') :5 5

C x y − +

+

=

24 9 5 3 171;

5 20 3 5 60D

+ −

⇒ � M 24 9 5 3 171

;5 20 3 5 60

D

− −

Page 28: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 27

24 9 5 3 171;

5 20 3 5 60

24 9 5 3 171;

5 20 3 5 60

f

f

e

e

= =

+ −

= − −

= −

'()&'()&'()&'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី៥

�. �����យ�បព<នQម� �� 2 2

(3 )( 3 ) 14

( ) 14( ) 36x

x y x y

x x y

xy

y y+ +

+ + =

+ =

�. �ក�គប"ប,- ច�ន�នប[ម p �./ម0� ! 112 2p − )ចក�ច"នDង 11p �

1. �គ !���� : ABC &នប,- �ម.uន 1 1 1, ,BBA CCA �បពBA� ��ង"ច�ន�ច G ( 1 1 1, ,A B C

4���2�5/ប,- �ជmង�ប"���� : ABC )� ប63 យ 1 1 1, ,BBA CCA �"�ងBង"\� Dក��]

���� : ABC =ម5��ប"��ង" 2 2 2, ,A B C �

��យបI% ក"J� 2 2 2 3GA GB GC

GA GB GC+ ≥+ �

E. �គ ! , ,a b c 'ប,- ច�ន�នព��# �ជ%&ន� បfg ញJ� 23

2 2 228

( )a b c ab bc ca

babc a c

+ + + + +

≥+ +

K. �គ ! 1007ច�ន�ច�ផpងA� �2�5/ប3ង"� ��យបI% ក"J &ន89ង��ច 2011ច�ន�ចក,- 5

�ផpងA� ព�ប,- គ*ច�ន�ច�ន�, ��/�ព5,).5&ន 2011ច�ន�ចក,- 5?

N. ប67 �"ព��ក��ង ន�ងព����]�ប"ម�� C �ប"���� : ABC �"ប67 �" AB ��ង" E ន�ង D �

��យប�ភ3�J �ប/ CE CD= �6� 2 2 24BCAC R+ = ( R ' ��ងBង"\� Dក��]���� :

ABC ) � '()&'()&'()&'()&

Page 29: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 28

ចំេល�យ

�. �����យ�បព<នQម� � 2 2

(1)

( )

(3 )( 3 ) 1

( 14 ) 3 (2)

4

6

x y

x y x xy

x xy

y

y

+ + + =

+ + =

=ង 0; 0x u y v= =≥ ≥ , �យ/ង;ន

( )4 2 2 4

6 4 2 2 4 6

10 3(2)

15 15

3 14(1

36),

uv u

u

u v v

u v u v v

+ +⇔

+ + + =

=

( )

5 3 3 5

6 4 2 2 4 6

6 5 4 2 3 3 2 4 5 6

6 5 4 2 3 3 2 4 5 6

6 6

66

10 3 14

15 15 36

6 15 20 15 6

6 15 20

3

36 2.14

36 2.14

( ) 22

2 2(

15

2

)

6

64

8 2

u

u

u

u v

v u v

u vu v

uv

u u v u v v

u v u v u v u v

u v u vu v

uv v

u v u v u v u v uv v

+ + =⇔

+ + +

+ =

− +

+ + =+ = ∨ − = − = −−

=

+ + + + + +⇔

− + − + − +

= =⇔ ⇔

= =

��យ ( 0,u v ≥ )

2

12

21

2

u

v

= +

= −

� M 2

12

21

2

u

v

= −

= +

.*ច�ន� �បព<នQ&នច��5/យ� 3 32; 2

2 2 + −

ន�ង 3 32; 2

2 2 − +

�. zប&J p 'ច�ន�ន).5��e#�ក, �យ/ង;ន� 112 (m (1)od2 )p p≡

Fន�#�-នj�ទD-�បទ Fermat ច��Z�ច�ន�នប[ម p , �យ/ង;ន� 11 112 (m d2 )op p≡ (2)

ព� (1) & (2)Sញ;ន p '��)ចក�ប" 11 2 2046 2.3. .312 11− = =

)� p 'ច�ន�នប[ម�6� { }2,3,11,31p ∈

+ ព�ន��! 2p =

�យ/ង&ន 22 2 (m )2 od 22≡ ,Sញ;ន 212 (mo1 11d ) (3)≡

មu9ង�ទ@� 10 1(mod112 ) (4)≡

Page 30: Vnmo 30 4-2011-grade 10

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Page 29

�6� ( )220 102 1(m 1)2 od1= ≡ , Sញ;ន 212 (m2 )od11≡ , ផ7�យនDង (3)�

.*ច�ន� 2p = ��e#�\5�

+ ព�ន��! 3p =

ព� (4)�យ/ង;ន ( )332 20 2.2 4 (mo 12 )2 d1= ≡ , .*ច�6� 322 1( 1)mod1≡/

Sញ;ន 322 1( 3)mod3≡/ ន�ង 322 2 ( 3)mod3≡/

.*ច�ន� 3p = ��e#�\5�

+ ព�ន��! 11p =

�យ/ង&ន ( )411 121 302 2 2 2 12 2p − = − = −

30 2 30 5 10 5 5 10 5) 1)(2 1)(2 1)(2 2 1)(2 12((2 )(2 2 1)+ + + − + − + += .

ជ�ន� 52 ��យ 32�O/យគ�បផp�នDង (4) , Sញ;នក��ងផ5គ�:�ន�&ន)�ក=- គ�:

52 1+ �ទ).5)ចកនDង 11;ន 3, �ប,- ក=- គ�:�ផpង�ទ@��ទQ)�)ចកម�ន�ច"នDង 11,

�6�ផ5គ�:)ចកម�ន�ច"នDង 11 121p = �

.*ច�ន� 11p = ��e#�\5�

+ ព�ន��! 31p =

�យ/ង��e#ព�ន��!�ម/5 3402 (m1 )od341≡ � M�ទ �

ព� (4)Sញ;ន 34340 102 12 (mod11) (5)= ≡

មu9ង�ទ@� 52 (m1 )od31≡ �6� 340 5 68(2 ) 1(mo 12 d 3 )= ≡ � គ�បផp�ច�ន�ច�ន�នDង (5)

Sញ;នបIg ��e#ព�ន��!គMព�� (11ន�ង 31�ទQ)�'ប,- ច�ន�នប[ម)

.*ច�ន� ច�ន�នប[ម).5��e#�កគM 31p = �

1. �យ/ង&ន� 2

1 1 2 1 1. .4

A A A B A Ca

AA = =2

1 2 4 aA

aA

m⇒ =

មu9ង�ទ@�� 2

2 1 1 21 3

3 4 3aa

a aGA m

mGA A A= + = + ≥ �6� 2 3

2 a

GA a

GA m≥

Page 31: Vnmo 30 4-2011-grade 10

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Page 30

.*ចA� ).� �យ/ង;ន� 2 3

2 b

GB b

GB m≥ ន�ង 2 3

2 c

GC c

GC m≥

�ព5�6� 2 2 2 3(1)

2 a b c

GA GB GC a a c

GA GB GC m m m

+ + + +

�យ/ង��e#��យបI% ក"J� 2 3a b c

a b c

m m m

+ +

ព��'.*ច�ន� 2

2 33.

4a aa

m a m+ ≥

� M 2

2 2 22 2 2

22 3

3. aa

ab c

aa m

m aa

b c+ + ≥ ⇔ ≥

+ +

�ធB/.*ចA� ).�, �យ/ង;ន� 2

2 2 2

2 3

a

b b

m a b c≥

+ + ន�ង

2

2 2 2

2 3

c

c c

m a b c≥

+ +

�ព5�6� 2 3a b c

a b c

m m m+ + ≥ (2)

ព� (1) & (2) �យ/ង;នបIg ��e#;ន��យបI% ក"�

Iw �n/�ក/�&ន ABC⇔ ∆ '���� :ម<ងp�

E. �គ ! , ,a b c 'ប,- ច�ន�នព��# �ជ%&ន� ��យបI% ក"J�

( )3 2

2 2 228

a b c ab bc ca

abc a b c

+ + + + +

≥+ +

Fន�#�-នj# �មiពក*��, �យ/ង;ន�

( ) ( ) ( ) ( )62 2 22 2 2 2 2

3 27

ab bc ca a a b ca

b cb cb bc ca a

+ + + + + + + + =

+ + ≤

+

Sញ;ន ( ) ( )

( )( )

33

2 2 2 2 62 2 2

27( ) ab bc caab bc ca ab bc ca

a ab bc ca a ab c b b cc

+ ++ + + +=+ + +

≥+ + ++ +

�យ/ង��e#��យបI% ក"J�

( ) ( )( )

( )63 2 2 23 2 6

512 2 4 2 4

5

35

27 ( )

25 5

( ) 27 ( )7

a b c ab bc ca ab bc ca

a

a b c

abbc a cc cb ab

+ + + + + +++ +

≥ ≥ =

B

A

1B

C 1A

1C

2A

2B

2C

G

Page 32: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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កL=ម# �មiពក*��, �យ/ង&ន� 3( )

23.2

27

3a b c

abc≥+ +

ប*កFងRនDងFងRប,- # �មiព�ង�5/�យ/ង;ន បIg ��e#;ន��យបI% ក"�

Iw " "= �ក/�&ន5���=)� 0a b c= = >

K. �យ/ង=ង Aន�ង B 'ព��ច�ន�ច&ន�ប)#ង)#ងប�ផ��ក��ងច��,ម 1007ច�ន�ច).5 !�

�យ/ងព�ន��!ប,- ច�ន�ចក,- 5�ង�� ម�ន� : ច�ន�ចក,- 5 M �ប" AB , ច�ន�ច

ក,- 5�ប" AX ).5 X 'ច�ន�ច,ម�យ).54���2ក��ង�ន��ច�ន�ច).5 ! (�ផpងព�

Aន�ង B ), ន�ងច�ន�ចក,- 5�ប" BX � �យ/ងនDង��យបI% ក"J ប,- ច�ន�ចក,- 5

�ន��ទQ)��ផpងA� �

ព��'.*ច�ន�, zប&J X ន�ង Y 'ព��ច�ន�ច,កL;ន �ផpងA� �O/យ�ផpងព� Aន�ង B �

ចt",").5ប,- ច�ន�ចក,- 5�ប" AX ន�ង AY ��e#�ផpងA� (�ប/ម�ន.*�\� ��ទ,

X ន�ង Y នDង��|�A� )� .*ចA� ).�, ប,- ច�ន�ចក,- 5�ប" BX ន�ង BY កL�ផpងព�A� ).�,

ច�ន�ចក,- 5 AX ម�នbច' M �ទ (��Z��ប/ម�នចDង�ទ, X នDង��|�នDង B ), កL.*ច�6�

).�, ច�ន�ចក,- 5�ប" BY ម�នbច'ច�ន�ច M �ទ� ច�ងប�a ប", �យ/ងzប&J N '

ច�ន�ចក,- 5��ម�ប" AX ន�ង BY � �ព5�6�, AYXB '�ប�5k*� ម, � M' AX , � M'

BY ).5��e#&ន�ប)#ង)#ង'ង AB , ច�ន�ច�ន�ម�នម�O��ផ5, ��Z��យ/ង;នzប&

JFង̀�" AB គM)#ងប�ផ���

.*ច�ន�, �យ/ង&ន89ង��ច 2011ច�ន�ចក,- 5�ផpងA� ព� 1007គ*ច�ន�ច).5;ន !�

ប)ន4ម�ទ@�, bច��@ប 1007ច�ន�ច�6��./ម0� !&ន��Dម 2011ច�ន�ចក,- 5�ផpងA� �

zSO�:j J, �2�5/F<កpច�ន�នព��, �យ/ង��ជ/យក 1007ច�ន�ច).54���2�5/ប,-

ក*F����ន 3, ..., 131, 20x = � �ព5�6�, &ន)� 2011ច�ន�ចក,- 5)�ប9��,c �).5

4���2�5/ប,- ក*F����ន .2 .,3 . ,, 4,5, 2012 �

N. �ប/ CE CD= �6����� : CDE )កងម;�

��ង" C (��Z�CD CE⊥ =ម5កq:�ប67 �"ព��

S�ងព���ប"ម��)�

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��ជ/យក�បព<នQក*F����ន.*ច�*ប (O 'ច�ន�ច

ក,- 5�ប" DE ), �យ/ង;ន�

( ;0), ( ;0), ((0;0) 0; ), ( ;0), ( ;0), A a B b C c DO c E c− .

=ម5កq:��ប"ប67 �"ព��ម���យ/ងSញ;ន�

2 2 2 2 2

2 2 2 2 2

( )

( )

AE AC AE AC c a a

EB CB EB CB b c b

c

c

+⇔ ⇔ − =− +

= =

2

2( ) 0)( aa ba

bc

b c− ⇔ =⇔ − = ,

.*ច�ន� ក*F����នច�ន�ច B គM 2

;0c

Ba

, .*ច�6�

( )24 2 2

2 2 2 2 22

(1)c a

Ac

BCa

cC a ca

+ =

+

+ = + +

=ង ( ; )I x y 'ផa���ប"�ងBង"\� Dក��]���� : ABC , �យ/ង;ន�

2 2 2 22 2

22 2 22 2

(( ))

( )

y x y cC

x aAI

ax cx a y

I

yIa

BAI

+ = + −=

⇔+ =

−− +

=

2 2

2 2

;22

aax

caa

y c

cc

I+ +⇔ ⇒

=

=

ន�ង 22 2 2 2

2 242

4 4 (2)a a

Ra

c

a

cCI

=

+ += =

ព� (1) & (2)Sញ;ន� 2 2 24BCAC R+ = (បIg ��e#;ន��យបI% ក") �

'()&

B A

C

D E x

y

O

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វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី៦ �. �����យម� �� 3000 3500 1500 1999 0x xx + + + =

�. �����យ�បព<នQម� �� 3 3

3 2

7

2

x

x y x

y − =

− + = −

1. ��យបI% ក"J &នពO�គ�:ម�យ�ប" 20112 ).5ម�ន&នផ7�ក�5ខ 0��� �

E. គ:6ផ5ប*ក ��Cន�គប"ប,- ��)ចកគ�"ធមn'�� �ប"ម�យZន"�ន�

K. �គ !���� : ABC \�Dកក��ង�ងBង" ( ; )O R � .DងJ�ប)#ងប67 �"ព��ក��ង ន�ងព����]�ប"

ម�� C �n/A� � បfg ញJ 2 2 24BCAC R+ = � '()&'()&'()&'()&

ចំេល�យ

�. Fន�#�-នj# �មiពក*��ច��Z� 3000x ន�ង 2999.ង�5ខ 1, �យ/ង;ន�

3000 2999 3000 | | 3000 (1)x xx + ≥ ≥ −

.*ចA� ).�, 3000 3 3999 1000 1000 (2)xx x+ ≥ ≥ −

ប*កFងRនDងFងRCន (1) & (2) ��ច)ចកនDង 2 , �យ/ង;ន�

3000 3500 1500 1999 0x xx + + + ≥ .

Iw �n/�ក/�&ន�ព5 1x = −

.*ច�ន� ម� �&នU)�ម�យគ�" 1x = − �

�. 3 3 3 3 3 3

3 2 3 2 2

7 7 8 1 (1)

2 5 ( 2)( 2) 1 (2)

x x x

x y x y y x y y

y y y

y x

− = − = − = −⇔ ⇔

− + = − − + = − + + = −

)���យ 2 2 0y y+ + > ��Z� 0∆ < , 6� !�

+ �ប/ 2y > �6� (1) 1

(2) 1x

x⇒

><

(ម�ន�ផ7GងH7 �")

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+ �ប/ 2y < �6� (1) 1

(2) 1x

x⇒

<>

(ម�ន�ផ7GងH7 �")

.*ច�ន� �បព<នQ&នច��5/យ)�ម�យគ�" ( 1; 2)x y= = �

1. 'ក")-ង 20112A = 'ពO�គ�:ម�យ�ប" 20112 ).5�5ខច�ងខ�ព� 0�

�ប/�5ខទ� m �ប" Aគ��ព��ង�- ��P�n/ 0ច��Z� m �*ចប�ផ��).5bចយក;ន

�6� 2011 11 2 .10mA A−= + 'ពO�គ�:ម�យ�ប" 20112 ).5 m �5ខច�ង�� យខ�

ព��5ខ 0�

�ធB/.*ច�ង�5/បន-ប67 ប"�P �6��យ/ងទទ�5;នច�ន�ន kA ម�យ).5'ពO�គ�:Cន 20112

).5 2011�5ខច�ង�� យខ�ព��5ខ 0�

ច�ន�នគ�"ធមn'��ប�ង̀/���យ 2011�5ខ�ប" kA 'ច�ន�ន).5�ផ7GងH7 �"�បoន�

E. ម�យZន"�ន 9 9 92 510 .==

��)ចកន�ម�យ��ប"ម�យZន"�ន&ន^ង� . (02 , 9)a bb a b≤ ≤

ផ5ប*ក ���ប"�គប"ប,- ��)ចក'ច�ន�នគ�"ធមn'���ប"ម�យZន"�នគM�

10 10

2 2

9 0 9 0 90 ,

4 1 25 1.5 4 . 25 .

4 1 25 12 a b a b

a b a b

A≤ ≤ ≤ ≤ ≤ ≤

− −= =− −

= ∑ ∑∑

.*ច�ន� ច��5/យ�ប"5�l�"គM� ( ) ( )10 104 2 15

72

1A =

− − �

K. =ង ,D E '�ជ/ងប,- ប67 �"ព��ក��ង ន�ងព����]ម�� C �

��យ CD CE= �6����� :CDE )កងម;���ង" C �

ព�ន��!�បព<នQក*F����ន Oxy ច��Z� Ox ��|�

នDង ,AB Oy ��|�នDងក�ព"ក�ព*5 C .*ច�*ប�

�យ/ង;ន� ( ;0), (0; ), ( ;0), ( ;0( ; ), )0 B b C c D Ea c cA − �

មu9ង�ទ@�� ( )( )

22 2 2 2

2 2 2 2 2

c aDA CA DA CA a

DB CB DB CB

c

bb c c

−= = =

+⇒ ⇒

+

B A

C

E D x

y

O

Page 36: Vnmo 30 4-2011-grade 10

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( )

22

24 2 22 2 2 2 2

2

) 0

(1

( )(

)

ca b c

a

c cAC a c

ab b

aBC

aac

+

⇒ − = ⇒ =

+⇒ + == + +

=ង ( ; )I x y 'ផa���ងBង"\� Dក��]���� : ABC , �យ/ង;ន�

2 2 2 22 2

2 222

2 2 22 2

( )( )

2( )

x aaAI

y x y ccCI

y yBI

xac

x a xAI y ca

+ = + − +=⇔ ⇔

+

− =

− −= + =

=

( )22 2

2 224

a cICR

a

+⇒ = = (2)

.*ច�ន� ព� (1) & (2) , �យ/ង;នបIg ��e#;ន��យបI% ក"� '()&'()&'()&'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី៧

�. �����យ#�ម� ��ង�� ម� ( ) ( )35 3 2 2 21 1x x x xx x x+ + ≤ + − +− �

�. ��យបI% ក"J ច��Z��គប"ច�ន�នគ�" ,a b �6� 5 5 30b aba − ⋮ �

1. �គ ! 8ច�ន�ច4���2ក��ង�ងBង"ម�យ'ម�យA� � បfg ញJ ក��ង 8ច�ន�ច).5 !គM&ន89 ង

��ច 2ច�ន�ច).5�ប)#ង�?ងព�ក?�*ច'ង � R �ប"�ងBង" �

E. �គ ! , ,a b c '�ប)#ងប,- �ជmង�ប"���� :ម�យ �O/យ S '�ក�Cផ7���� :�

បfg ញJ 4 4 4 4b Sa c+ + ≥ , ក��ងក�:� ,).5Iw មiព�ក/�&ន?

K. ��/&នប9�6n ន��ប@បក��ង ���@បប,- ច�ន�ន 31, 41, 51, 61,21, 71, 81 �./ម0� !ផ5ប*ក

�ប"ប�នច�ន�ន�A� )�ង)ចក�ច"នDង 3 ?

N. �គ !���� : ABC , �ក�ន�� ប,- ច�ន�ច M �./ម0� !�

2 2 2 2 2 2; 2). ).a MB bMC MA MC MAMB+ = + = � '('('('()&)&)&)&

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ចំេល�យ

�. ( ) ( )35 3 2 2 21 1 (*)x x xxx x x−+ + ≤ + − +

5កq:r ក�:�"� 0x ≥

+ ក�:� ទ��: 0x =

(*) 0 1⇔ ≤ (ព��'ន�ចa)

.*ច�ន� 0x = 'Uម�យ�ប"# �ម� ��

+ ក�:� ទ��: 0x >

)ចកFងRS�ងព��Cន# �ម� �នDង ( )2 2 01x x + > �

�យ/ង;ន� ( )( )

( )( )

( )32 2 25 3

2 2 2 2 2 2

1 1

1 1 1

x x xx

x x x x x x

xx x + − ++ + ≤+

−+ +

( )( )

( ) ( )( )

( )

24 2 2 2

22 2 2

2

22

2

11

1 1 11

1

1 1 1 11

1

1 1 1

1 1

1

11

1

1

x x x x

xx x x

x xx

x xx x

xx

x xx xx

xx xx x

x x

x x

x

x

+ − −

+ − − −

+ + + − +⇔ ≤

+ +

+⇔ ≤++

⇔ ≤ ++ +

+ − −+

+ −+

⇔ ≤

=ង ( )1

2t x tx

≥= +

�ព5�6�, # �ម� ��P' 1 11tt

t t≤ −− −

211 1 1 1 1

1 2t t tt t t t

t t tt

≤ − ⇔ − −≤− −− −+

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��យFងRS�ងព���ប"# �ម� ��ទQ)�# �ជ%&ន�6�

2

01 1 1

0 2 1 1t t tt t t

t≤ − ⇔ ≤

− − + − −

(ព��'ន�ចa 2t∀ ≥ )

.*ច�ន� ច��5/យ�ប"# �ម� �គM 0x ≥ �

�. �យ/ង&ន� 5 5 5 5b ab a b ab aba ab− = + − − ( ) ( )4 41 1ab a ab b= −− −

( )( ) ( )( )( )( ) ( )( )

( )

2 2 2 2

2 2 2 2

2 2

1 1 1 1

1 4 5 4 5 1

4)( 1)( 1)( 5 ( 1

ab a a ab b b

ab a a ab b b

ab a a a a ab

= −

=

− + +

= − +

− − + − + −

− + − −

( )2( 1)( 1) 5 ( 1)( 1)4ab b b b ab b b−− − + + − +

( ) ( )2 2

( 2)( 1)( 1)( 2) ( 2)( 1)( 1)( 2

1

)

15 5

ab a a a a ab b b b b

ab a ab b

= − − + + − − − + +

−−+ −

+

��យ ( 2)( 1)( 1)( 2)ab a a a a− − + + ន�ង ( 2)( 1)( 1)( 2)ab b b b b− − + + )ចក�ច"នDង 2,3

ន�ង 5�

.*ចA� ).� ( )25 5 . ( 1)( 1)1ab a b a a a= − +− )ចក�ច"នDង 2,3,5 ន�ង 2 1)5 (ab b − )ចក�ច"

នDង 2,3,5 ,មu9ង�ទ@� (2;3) (3;5) (2;5) 1= = =

�6� 5 5 2.3.5ba ab− ⋮ � M 5 5 30b aba − ⋮ �

1. ក��ងច��,ម 8ច�ន�ច).5 !, &ន89 ង��ច 7ច�ន�ចម�ន��|�A� នDងផa�� O �ប"�ងBង"�

ច�ន�ចន�ម�យ�).5 ! �;ក.'4���2�5/ �ម�យ�ប"�ងBង" ( )O �

+ �ប/&ន 2ច�ន�ច (ក��ងច��,ម 8ច�ន�ច).5 !) 4���2�5/ �)�ម�យ�6��ប)#ង�?ង

ព��ច�ន�ច�ន� គMខ3�'ង � R �

+ �ប/ �ន�ម�� �"=ម)� 1ច�ន�ច)�ប9��,c � �6�&ន89 ង��ច 7 �ក�:�";ន 7ម���2

ផa��ក��ងច��,ម�6�&ន89 ង��ចម�� 1).5 0 01.360 0

76≤ < �

�យ/ងbចzប&J� � 060 (*)AOB < , ).5 Aន�ង B 'ច�ន�ច 2ក��ងច��,ម8ច�ន�ច

).5 !ម�ន��|�A� នDងផa�� O �

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ព� (*) Sញ;ន � M � 060OAB > � M � 060OBA > �

�ប/ � 060OAB > �6� BB RO OA A⇒ <>

�ប/ � 060OBA > �6� BA RO AB A⇒ <>

.*ច�ន� ក��ង 8ច�ន�ច).5 !4���2ក��ង�ងBង" ( ; )O R )�ម�យ &ន89 ង��ច 2ច�ន�ច).5

�ប)#ង�?ងព�ក?ខ3�'ង � R �

E. =ម�*បមន-គ:6�ក�Cផ7�O��ង�

( )( )( )2 2 2 2

a b c a b c a c b b c aS p p a p b p c

+ + + − + − + − = − − − =

(4 )( )( )( )a b c a b c a c b bS c a⇔ = + + + − + − + −

Fន�#�-នj# �មiពក*��ច��Z�ប�ច�ន�ន# �ជ%&ន , ,a c b bb aa cc+ − + − + − , �យ/ង;ន�

( ) ( )3( )( )( ) .

27

a b ca b c a c b b c a a b c

+ ++ − + − + − + +≤

( ) ( ) ( ) ( ) ( )

( ) ( )

2

22

2 2 2

3 3

4 12 33 3

12 3 2 2 2

a b ca b c a b

a b c a

c a c b b c a

a b cS S a

b cS b a

b

c

c

+ ++ +⇔ ≤

⇔ ≤ ⇔ ≤

⇔ ≤ + + +

+ − + − + −

+ ++ +

+ +

បន-Fន�#�-នj# �មiពក*��ច��Z�គ*ច�ន�ន# �ជ%&ន 2 2 2 2 2 2, ; , ; ,ca b b c a �យ/ង;ន

2 2 2 2 2 22 ; 2 ; 2b aba b cc bc a ac+ ≥ + ≥ + ≥

2 2 212 23 2 2a b c ab bS c ca⇔ ≤ + + + + +

( ) ( ) ( )2 2 2 2 2 2 2 2 2a b c b cc aa b≤ + + + ++ +++

( ) ( )2 2 2 2 2 212 3 343 b cS a S b ca⇔ ≤ + + ⇔ ≤ + +

Fន�#�-នj# �មiព Bunhiakovski �យ/ង;ន�

( ) ( )2 2 2 4 4 4

2 2 2 4 4 4

1 1 1

3

b c b c

b c b c

a a

a a

+ +⇔ + + ≤ + +

⇔ + + ≤ + +

Page 40: Vnmo 30 4-2011-grade 10

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6� ! ( )2 2 2 4 4 4 4 4 444 3 3b c b ca S bS a a c≤ + + ≤ + + ⇔ ≤ + +

Iw មiព�ក/�&ន5���=)� a b c= = � M���� :�6�'���� :ម<ងp�

K. ��យ 0 3); 1(mod 3); 2 (mod 321 (m );od 31 41 5 0 (mod 3)1≡ ≡ ≡ ≡

1(mod 3); 71 2 (mod 3); 81 061 ( 3)mod≡ ≡ ≡ .

�6� �យ/ងbច����k/ង# �ញប,- ច�ន�ន 31, 41, 51, 61,21, 71, 81��យប,- ច�ន�ន

1, 2, 00, ,1, 2, 0� zប&J 1 2 3 4 5 6 7, , , , , ,a a a a a aa ' ���@បម�យ).5�ផ7GងH7 �"

��:/ ��បoន�

�យ/ង�ឃ/ញJ� 1 2 3 4 4 5 6 7( ) ( )0 a a a a a a a a≡ + + + + + + + ≡

1 2 3 4 5 6 7 4 4) (mo )( d 3a a a a a aa a a≡ + + + + + + + ≡ .

��យ 4 0 (mod 3)a ≡ �6� 1 2 3, ,a a a ��e#)�' ���@បម�យ�ប"ប,- ច�ន�ន 1.2.3�

.*ច�6�, 1 2 3 1 2 3 40 (mo )d 3a a aa a a a≡ + + ≡ + + +

��យ 1 2 3 4 4 5 6 7 (mod0 3)a a a a aa a a+ + + ≡ + + + ≡ , �6� 1 5 ( d3)moaa ≡

.*ចA� ).�, �យ/ងកL��យបI% ក";ន 5��ប"�ប"ប,- ច�ន�ន 1 2 3, ,a a a កL��e#;ន

ក�:�";ន)�ម�យគ�"គM��យប,- ច�ន�ន 1 2 3, ,a a a �

.*ច�ន� &ន 3.33.2 ! 144= ��ប@ប��@ប�

N. 2 2 2 2 2 2 0). MC MA MC MM B Aa B M+ = ⇔ + − =

=ង E 'ច�ន�ចក�:�"��យ� 0EB EC EA+ − =� � � �

( E 'ក�ព*5ទ�ប�ន�ប"�ប�5k*� ម ABEC )

�យ/ង;ន� 2 2 2 2 2 2 2MC MA ME EB EC EB AM + − = + + −

( ) ( )22 2 2

2 2 22 . 2 . 2 . .cos

EC EB EC

EB EC ME AB

ME E

AC ME AB AM C

B

E A

= −

=

+ + +

− = − = −

� �

� � � �

.*ច�ន�,

+ �ប/ម�� A'ម��S5 �6��ន��ច�ន�ចប,- ច�ន�ច M '�ន��ទ�ទ�

+ �ប/ម�� A'ម��)កង �6��ន��ច�ន�ចប,- ច�ន�ច M គM { }E �

Page 41: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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+ �ប/ម�� A'ម���|ច �6��ន��ច�ន�ចប,- ច�ន�ច M '�ងBង" ( ); 2 . .cosE AB AC A �

).b =ង I 'ច�ន�ចក,- 5�ជmង BC �

2

2 2 2 2 222

2 2MB

CC

MB MAM MIA+ = ⇔ + =

( )( )2 2

2 2

4 4MI MA MI MA

BC BM

CMA I =⇔ − = ⇔ + −

� � � �

2 2

24

.4

.MJ IA JHBC BC

AI⇔ = ⇔ =� � � �

( J 'ច�ន�ចក,- 5�ប" ,AI H 'ច��,5)កង�ប" M �P�5/ AI )�

2

8

BCJH

AI⇔ =

.*ច�ន� �ន��ច�ន�ច�ប"ប,- ច�ន�ច M 'ប67 �")កងនDង AI ��ង" ,H ក�:�"��យ

2

8

BCJH

AI= �

'()&'()&'()&'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី៨

�. �����យ�បព<នQម� �� 2 2

2

2 2 3 0

2 2 7 0

b a b

a ab b

a − − + + =

− + + =

�. �ក *n∈ℕ �./ម0� !ច�ន�ន)ផ�ក�ប"ប3ង").5��e#;ន)ចក')ផ�ក��យ n ប67 �" �"A�

ព�ម�យ�Pម�យ �O/យម�ន&នប�ប67 �",�បពBA� , 'ច�ន�ន ���;ក.ម�យធ�'ង

100ន�ង�*ច'ង 900�

1. �គ !���� : ABC ម�នម;�, �ងBង"\� Dកក��ងផa�� I �ប"���� : ABC ប9�នDង

ប,- �ជmង , ,BC CA AB ��@ងA� ��ង"ប,- ច�ន�ច ', ', 'A B C � =ង ' ' 'M A I B C= ∩ ,

ប63 យ AM �" BC ��ង" Q � គ:6ផ5�ធ@ប�ក�Cផ7���� :S�ងព�� ABQ

ន�ង ACQ �

Page 42: Vnmo 30 4-2011-grade 10

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E. �គ ! 3ច�ន�នព��# �ជ%&ន ,,a b c �ផ7GងH7 �"5កq:r � 2 2 2 1ba c+ + =

�ក��C5�*ចប�ផ���ប"ក�នyម 2 2 2 2 2 2

a b cP

b a ac c b+ ++

+= + �

K. ��Iw �ជ/ង)ក�ក 'Iw ).5��ម&នប67 �"ព��)ខBងA� �ប" ���ជmង�n/ 1�ក=�

�គ�ក" 200Iw �ជ/ង)ក�កច*5�Pក��ង�ងBង"ម�យ&ន ��n/ 5�ក=� ��យបI% ក"J

&ន89ង��ច Iw �ជ/ង)ក�កព�� �"A� �

N. �គ ! ABC∆ ម;���ង" A� ប67 �" AC &នម� � 3 5 0x y− − = �=ង H '

ច�ន�ចក,- 5�ប" BC � D 'ច��,5)កង�ប" H �P�5/�ជmង ,AC M 'ច�ន�ច

ក,- 5�ប" HD � ប67 �" BD �"=ម (8; 5)E − � AM &នម� ��

11 7 5 0x y− − = � ច*����ម� ��ជmង ,AB BC � '()&'()&'()&'()&

ចំេល�យ

�. ����បព<នQ�k/ង# �ញ� 2 2

2 2

4 ( 1) 1

( ) 9 ( 1

)

1

( 1

)

b

a b b

a + = − +

+ = −

+

ព�ន��!�បព<នQ����យ Oxy �

=ង ( ;3(1 ),;1), ( ;0)B a CA b ,

������យ�បព<នQម� � 3 យ�P' ��ក ,B C

�./ម0� !���� : ABC '���� :ម<ងp�

�ប/ 1b > , �ងBង"\� Dក��]���� : ABC �" Ox ��ង" D �

�ព5�6� � �0 0180 120ADB ABC= − = �6�ម� � AD គM� 3( 1) 1y x= − − +

D AD Ox= ∩ , Sញ;ន 3 1;0

3D +

� � 060BDC BAC= = , ប67 �" BD ប�ង̀/�'ម�យ Ox ;នម�� 060 �6�ម� �

3 13

3:EB y x

+= −

, Sញ;ន 4 3;3

3B +

, ជ�ន�ច*5�បព<នQម� �

B

A

C D x

y

Page 43: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 42

).5 !�យ/ង;ន 5 3

3b

+=

�ប/ 1b ≤ , .*ចA� នDងក�:� �ង�5/).� �6�គM&នច�ន�ច C )�ម�យគ�"�ផ7GងH7 �"5កq:r

ABC '���� :ម<ងp� =ង ( )d 'ប67 �" �"=ម A �O/យ�បនDង Oy �

�d ', 'B C ឆ3��នDង ,B C �ធ@បនDង ( )d � �ព5�6����� : ' 'AB C ម<ងp�

.*ច�ន� ក��ងក�:� �ន� 3 4 3 5,

3 3a b =− −=

.*ច�ន� �បព<នQ&នច��5/យគM� 4 3 5 3 3 4 3 5( , ) ; ; ;

3 3 3 3a b

+ + − −=

�. �ងqបច��5/យ�

+ ច�ន�ន)ផ�កប3ង").5��e#;ន)ចក')ផ�ក ��យ n ប67 �" �"A� ព�ម�យ�Pម�យ �O/យ

ម�ន&នប67 �"ប�,�បពBA� គM�

2( 1) 2

..2 .2

2 3 12

n n n nn+ + + + + ++ = + =

+ �យ/ង�កច�ន�នគ�"# �ជ%&ន [ ]3010;y ∈ �./ម0� !�

2

2 2 222

22 0

ny n

nn y

+ + ⇔ + + − ==

( ) ( ) ( )2 2 278 7 (2 1 2 11 2 1) y x y yy xx⇒ ∆ = − = + ⇒ = + + + −−

2 1 7x y m+ − =⇒ � M 2 1 7 ( )x y m m+ = ∈+ ℕ

+ ក�:� ទ��: 2 1 7x y m+ − =

2 2 2 2 2 22 1 7 0 8 1 8 1my m m k k my⇒ − − − = ⇒ + = ⇒ − = .

(ម� � Pell ) ⇒ U'ច�ន�នគ�"# �ជ%&ន�*ចប�ផ��គM (3;1)

4

5

y

n

= =

(�\5)

Uទ�ព��គM 23(17;6)

32

y

n

==

(យក)

+ ក�:� ទ��: 2 22 1 2 7 07 1x y m y my m⇒ + − −+ = =+

Page 44: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 43

2 28 1mk⇒ − = ⇒ U'ច�ន�នគ�"# �ជ%&ន�*ចប�ផ��គM (3;1) 2

2

y

n

= =

(�\5)

Uទ�ព��គM� 11(17;6)

15

y

n

==

(យក)

ន���h ន� 15n = � M 32n = �

1. =ម Aគ*ប67 �" ||d BC , =ង ' ', dS IN B C Ad ′== ∩ ∩

' , ', 'A S Sd B C⇒ ⊥ ⇒ 4���2�5/�ងBង"Fង̀�"ផa�� AI , ��យ ' 'IB IC SM= ⇒

'ប67 �"ព��ក��ង�ប" �B SC′ ′ ⇒ 'ប67 �"ព����] �B SC′ ′

, , ,N M B C′ ′⇒ 'ប�):កbម9*ន�ច

(1)NB MB NB NC

NC MC NC MC

′ ′ ′ ′⇔ ⇔

′ ′ ′ ′−= = −

=ម M ង"ប67 �"�បនDង AN , ប67 �"�ន�

�" ',AB AC′ ��@ងA� ��ង" E ន�ង F �

Fន�#�-នj�ទD-�បទ=)5, �យ/ង;ន� (2); (3)B M ME C M FM

B N AN C N AN=

′′ ′

= −′

ព� (1), (2), (3)ME FM

ME FMAN AN

ME FM= = ⇔ =⇒ ⇒

��យ || 1ABQ

ACQ

SFE BC QB QC

S⇒= =⇒

E. �យ/ង&ន� ( )3

3 3 21 1 21

273 3 3 3 3 33

aa a a a+ ≥ → ≤+ = −

( )2

22

1 3 3 3 3

2 21(

11)

a a

aa a−≥ → ≥

−→

.*ចA� ).�� 2 2

2 2(2);

3 3 3 3

2 21 1(3)

b b c c

b c≥

− −≥

ប*កFងRនDងFងRនDង (1), (2), (3):

( )2 2 22 2 2

3 3

21 1 1

a b ca

a bb c

c+ +

− − −≥ + +

A S N

B 'A Q C

'C

E

'B

F

I

M

Page 45: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 44

Sញ;ន 2 2 2 2 2 2

3 3

2c c

a bP

b b

c

a a= + ≥

+ ++

+

3 3 1

2 3a cp b⇔ = = ==

.*ច�ន� 3 3 1min( )

2 3P a b c⇔ = = == �

K. ច��Z��*ប�ជ/ង)ក�កន�ម�យ� �យ/ងព�ន��!�ងBង").5&នផa����ង"ផa���*ប�ជ/ង)ក�ក ន�ង&ន

��n/ 1

2 2 �

�យ/ង��យបI% ក"J �ប/�ងBង"ព���6� �"A� �6��*ប�ជ/ង)ក�កS�ងព��នDង �"A� ).��

ព��'.*ច�ន�, �យ/ង&ន�ប)#ង�?ងផa��S�ងព���ប"�ងBង" �"A� គMម�ន)#ង'ងព��.ងCន

�ប)#ង ��ប"ព�ក?�ទ, .*ច�6� �ប)#ង�?ងផa���ប"�*ប�ជ/ង)ក�ក).5��e#A� �ប"ព�ក?

ម�ន)#ង'ង 1

2 �ទ�

ព�ន��!�*ប�ជ/ង)ក�កក�:�"��យប,- ប,- �3 ប�ប"�*ប�ជ/ង)ក�កទ�ម�យ ន�ងផa���ប"

�*ប�ជ/ង)ក�កទ�ព���

��យ�ប)#ង�ប"�3 ប�n/ 1

2,��ប)#ងFង̀�"�ទeង�ប"ច��� :)កងម�ន)#ង'ង 1

2

�6�នDង&ន�3 បម�យ�ប"�*ប�ជ/ង)ក�កទ�ព�� �"ច��� :)កង�6�, 6� ! ?នDង �"

�*ប�ជ/ង)ក�កទ�ម�យ�

.*ច�ន� �ប/�ងBង"S�ងព�� �"A� �6��*ប�ជ/ង)ក�កS�ងព��កL �"A� ).��

ព�ន��!�ម/5�ងBង").5&ន ��n/ 5�ក=� �ក�Cផ7�ងBង" 25π , �ងBង"ន�ម�យ�).5&ន

� 1

2 2&ន�ក�Cផ7 1

8π �

�យ/ង�ក"ច*5�Pក��ង�ងBង"&ន � 5�ក=ន*#�ងBង").5&ន � 1

2 2ច�ន�ន 200�ងBង"�

zប&J ប,- �ងBង"�ន�ម�ន �"A� , �យ/ង;នផ5ប*ក�ក�Cផ7�ប"ព�ក?�n/នDង

�ក�Cផ7�ងBង"ធ�� មu9ង�ទ@� ប,- �ងBង"�2�ច"ព�A� ម�នbច�គបជ��ន*#�ងBង"ធ�;ន�ទ�

Page 46: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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.*ច�6� ផ5ប*ក�ក�Cផ7ប,- �ងBង"�*ច��e#�*ច'ង�ក�Cផ7�ងBង"ធ�, (ផ7�យA� )�

.*ច�ន� ��e#&ន89ង��ច�ងBង"ព�� �"A� �

Sញ;ន &ន89 ង��ច�*ប�ជ/ង)ក�កព�� �"A� �

N. A'ច�ន�ច�បពB�ប" AM នDង AC �6�ក*F����នច�ន�ច A �ផ7GងH7 �"�បព<នQ�

3 5 0 3(3;4)

11 7 5 0 4

x y xA

x y y

− − = = ⇔

⇔− − = =

=ង K 'ច��,5)កង�ប" B �P�5/ AC �

�ឃ/ញJ HD ';�មធ!ម�ប"

BCK∆ �6� D 'ច�ន�ចក,- 5�ប" KC �

ង" Ax �បនDង HD , By �បនDង AC ,

�ព5�6��យ/ង;ន , , ) 1, ,( , ( , , ) 1AH BCAD AM Ax BK BD By= − = − �

)��យ/ង�ឃ/ញJ AH BC⊥ , AD BK⊥ , Ax By⊥ , Sញ;ន AM BD⊥ �

BD �"=ម E �O/យ)កងនDង AM �6�&នម� �� 7 11 1 0x y+ − =

D 'ច�ន�ច�បពB�ប" BD នDង AC �6�ក*F����ន D �ផ7GងH7 �"�

73 5 0 7 45 ;7 11 1 0 4 5 5

5

xx y

Dx y

y

⇔ ⇒

=− − = − + − = = −

ប67 �" HD �"=ម D �O/យ)កងនDង AC &នម� �� 3 1 0x y+ + =

M 'ច�ន�ច�បពB�ប" HD នDង AM � ក*F����ន M �ផ7GងH7 �"�បព<នQ�

13 1 0 1 25 ;

11 7 5 0 2 5 5

5

xx y

x yy

M

=+ + = − − − = = −

M 'ច�ន�ចក,- 5�ប" ( 1;0)HD H −⇒

ប67 �" BC �"=ម H �O/យ)កងនDង AH &នម� �� 1 0x y+ + =

B 'ច�ន�ច�បពB�ប" BC នDង BD �6�ក*F����នច�ន�ច B �ផ7GងH7 �"�បព<នQ�

B

A

C

D H

M

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 46

1 0 3

7( 3;2

11 1 0)

2B

x y x

x y y

+ + = = − + − = =

⇔ −

ប67 �" AB �"=មព��ច�ន�ច ,A B &នម� � 3 9 0x y− + = � '()&'()&'()&'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី៩

�. �គ ! ,,a b c'ប�ច�ន�នព��ខ�ព� 089ង, ! 0ab bc ca+ + ≥ �

��យបI% ក"J� 2 2 2 2 2 2

1

2b c

ab bc ca

a b c a+

+ ++ > −

+ �

�. �ក�គប"ច�ន�នគ�"# �ជ%&ន n �./ម0� ! 9 16n + ន�ង 16 9n + �ទQ)�'ច�ន�ន ���;ក.�

1. �គ !���� : ABC ម;�).5 AB AC= � =ង ,O I ��@ងA� 'ផa���ងBង"\� Dក

��]នDង\� Dកក��ង���� : ABC � ច�ន�ច D 4���2�5/�ជmង AC 89ង, ! ID

�បនDង AB � បfg ញJ CI OD⊥ �

E. ក��ងប3ង"�ប�ប"��យ����យ)កង Oxy �គ !���� : ABC ).5 ( 1;5) ; ,, ( 3 1)BA − −−

(7; 1)C − � �2�5/ប,- �ជmង , ,BC CA AB=ម5��ប"�គ�dប,- ច�ន�ច , ,I J K �

ក�:�"ក*F����ន�ប" , ,I J K �./ម0� !ប� �&������ : IJK &ន��C5�*ចប�ផ���

K. ��យបI% ក"J ច��Z�ច�ន�នគ�"ធមn'�� n , &នច�ន�នគ�"ធមn'�� p 89ង, !�

( )2011 2010 1n

p p+ = + − �

N. �����យ�បព<នQម� ��

4 4

4 2 2 42 2

(1)

1

121 12

22 12114 (2)

2

4y

x yx x y y

x yx

x

x y

y

− − =

++

=+

+

'()&'()&'()&'()&

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 47

ចំេល�យ

�. �យ/ង&ន� 2 2 2 2 2 2

ab bc ca

a cb acb+ + ++ + =

( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )

2 2 2 2 2 2

2 2 2

2 2 2 2 2 2

2 2 2

2 2 2 2 2 2 2 2 2

2 2 2

2 2 2

2 2 2 2 2 2

1 1 1 3

2 2 2 2

( ) ( ) ( ) 3

22 2 2

( ) ( ) ( ) 3

22 2 2

2 2 3

22

31

2

b c a

b c a

b c b c

ab bc ca

a b c

a b b c c a

a b c

a b b c c a

a a a

a ab bc ca

a

ab bc ca ab bc ca

a

b c

b c

b c

b c b ca

= + + + + + −

+ + += + + −

+ + +> + + −

+ +

+ + +

+

+= −

+ + + +

+

= + −

+

+ + + + + +

+ +

+ +

+ + + += 1 1

2 2− ≥ −

�. �ប/ 9 16n + ន�ង 16 9n + �ទQ)�'ច�ន�ន ���;ក.�6� 29 6n a+ = ន�ង

216 9n b+ = ច��Z� ,a b 'ព��ច�ន�នគ�",កL;ន�

�ព5�6� 2(9 16)(16 9) ( )n n ab+ + = កL' ���;ក.).�

=ង ( )2 2 2(9 16)(16 9 144 9) 14416nT n nn n= + + = ++ +

( )2 2 2 2(1 12 ) 9 126n n+ += +

�យ/ង;ន ( )2 2 2 2 2 2(12 ) 16 (12 15)(12 12) 9 12n nn n< + + < ++ +

.*ច�ន� 2(12 13)n nT = + � M 2(12 14)n nT +=

( )2 2 2 2 2(12 13) (1). 9 22 1 1) 6na T n nn= + = + + + � M 312 169 337 144n n+ = +

Sញ;ន 1n =

�ព5 1n = �6� 29 16 16 9 25 5n n+ = + = =

( )2 2 2 2(12 14) (1). 9 442 ) 16 1nT n nnb = + ++ = + � M 336 196 337 144n n+ = +

Sញ;ន 52n =

Page 49: Vnmo 30 4-2011-grade 10

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Page 48

�ព5 52n = �6� 29 16 484 22n + = = ន�ង 216 9 841 29n + = = �

1. �យ/ង&ន, ���� : ABC ម;���ង" A �6� , ,A O I 4���2�5/�ម.uទ<� AP

�ប" BC 'ម�យA� � =ង E 'ច�ន�ច�បពB�ប" ID នDង ,BC F 'ច�ន�ច�បពB

�ប" ODនDង ,BC Q 'ច�ន�ច�បពB�ប" CI នDង DO �

).a ក�:� � 060BAC <

ក��ងក�:� �ន� O �2ច�63 � Aន�ង ,I Q

4���2ក��ង���� : PAB �

�យ/ង;ន� ��យ ||DE AB

�6� � � � �1

2CDI CAB COB COI= = =

(��Z� O 'ផa���ងBង"\� Dក��]���� :)

.*ច�ន� ច��� : CDOI \�Dកក��ង;ន, � � �0180ICD IOD QOI= − =

Sញ;ន � � �( ) � �( )0 0180180CQD QOI QIO ICD PIC= − + += −

� �( )0180 ICP PCI−= +

(��Z� I 'ផa���ងBង"\� Dកក��ង���� :�6� � �ICD ICP= )

).b ក�:� � 060BAC > :

ក��ងក�:� �ន� I �2ច�63 � O ន�ង ,A Qន�ង C �2�ង)�ម�យឈមនDង AP �

�យ/ង;ន� � � � � 0180IDC IOC BAC AOC+ = + =

Sញ;ន ច��� : DIOC \�Dកក��ង;ន�

.*ច�6� �យ/ង;ន�

� � �( ) � �( )0 0180180CQD DCQ QDC QCP ODC= − + += −

� �( ) � �( )0 0 0180 180 90QCP OIC ICP PIC+= −= −− +

).c ក�:� � 060BAC = :

�យ/ង;ន I ��|�នDង O �O/យ DF ��|�នDង DE �ពមS�ង�បនDង AB ន�ង CI

D

A

C B

O

I Q

E F

P

O

B

A

C

D

E

F

P

Q I

Page 50: Vnmo 30 4-2011-grade 10

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Page 49

)កងនDង ODគM' �ព���

E. =ង ,P Q'ច�ន�ចឆ3��នDង I �ធ@បនDង ,AB AC � �ព5�6� ប� �&���ប"���� : IJK

�n/នDង PK KJ IQ+ + � ! I �2នDង�

PK KJ IQ+ + ខ3�ប�ផ���ព5 , , ,P K J Q ��"��ង"ជ��A�

�យ/ង;ន ( 2;6), (8; 6), . 20 0AB AC AB AC= − = − = >� � � �

�6�ម�� BAC = α'ម���|ច�

មu9ង�ទ@� �យ/ង&នម�� 01802PAQ α <= (ម�ន)�ប�ប|5)

.*ច�6� PQ �"ប,- �ជmង ,AB AC ��ង" ,M N � �d K ��|�A� នDង ,M J ��|�នDង

N , �យ/ង;ន� 2 2 2 22 . cos 2 (1 cos 22 )AP AQPQ AP AQ AI= + − α −= α

Sញ;ន� PQ ខ3�ប�ផ���ព5 AI ខ3�ប�ផ��� �ព5�6� I ��|�A� នDង�ជ/ងក�ព" 'A

ង"ព� A �O/យ ,M N ��|�នDង�ជ/ងក�ព" ', 'C B ង"ព� ,C B�

�កក*F����ន �កក*F����ន �កក*F����ន �កក*F����ន ,',A B C′ ′ :

ចt"," '( 1; 1)A − −

ម� �ប67 �" : 6 2 16 0xA yB − + =

ម� �ប67 �" 6 0': 2 8xC yC + + =

ក*F����នច�ន�ច : ( 2;2)' CC ′ −

ម� �ប67 �" :3 4 17 0xA yC + − =

ម� �ប67 �" 3 0': 4 9xB yB − + =

ក*F����នច�ន�ច ( )' 3 / 5;: 19 / 5B B′ �

K. =ង 2011m = �6��យ/ង;ន ( )1 1n

m m p p=+ − + −

=ម�*បមន- Newton �យ/ង;ន�

( ) ( ) ( ) ( ) ( )0 1 10 11 1 1n n n

n nm m m mC C m m−

+− +− −=−

( ) ( )0... 1

nnnC m m+ + −

).a �ប/ n គ*� ផ-��ប,- ��

�យ/ង;ន� ( ) 1 (1)1 ( )n

n nA Bm m m m= ++ − −

I 2−

2 4

H

6 A

2 'C

B C

M

K

Q

0 2− 4− 6−

4

H

6 8

P

N

'B J

'A

Page 51: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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).5 ,n nA B 'ប,- ច�ន�នគ�"ធមn'��

.*ចA� ).�� ( ) 1 (2)1 ( )n

n nA Bm m m m= −− − −

គ�:FងRនDងFងR�យ/ង;ន� 2 2 ( 1) (1 3)n nB mA m− −=

=ង 2np A= �6�ព� (3)�យ/ង;ន 2 2( 1) 1n nmB m A− = −

Sញ;ន 2( 1) 11n nB m m A p−− = = −

.*ច�ន� (1)����k/ង# �ញ;ន� ( )1 1n

m m p p=+ − + −

).b �ប/ n 'ច�ន�ន�� ផ-��ប,- ��=ម m ន�ង 1m − �យ/ង;ន�

( ) ( )1 41n

n nm m m DC m+ −=+ −

ន�ង ( ) ( )1 51n

n nm m m DC m− −=− −

ច��Z� ,n nC D 'ប,- ច�ន�នគ�"ធមn'��

គ�:FងRនDងFងRCន (4), (5)�យ/ង;ន� 2 2(1 1)n nmC Dm − −=

=ង 2np mC= �6�ព� (4)�យ/ង;ន 2( 11) nD pm − = −

ជ�ន�ច*5 (4)�យ/ង;ន� ( )1 1n

m m p p=+ − + −

N. 5កq:r � 0xy ≠

�ប/ x y= ± �6� (1)ម�នម�O��ផ5�

�យ/ង;ន (1)មម*5នDង� ( )4 44 12 (31 2 )1 2xy x xy y= −−

�O/យ (2)មម*5នDង� ( ) ( )4 2 2 4 2 214 (4122 121 )x y y yx x x y+ + + = +

គ�: (4)នDង ( )x y− �យ/ង;ន�

( ) ( ) ( ) ( ) ( )4 2 2 4 2 214 (122 5121 )x x x y yy x y xx y y − = + −+ + +

គ�: (3)នDង ( )x y+ �យ/ង;ន�

( )( ) ( )( )4 44 121 122 (6)xy x x y xy y x y+ = − +−

យក (5).ក (6)FងRនDងFងR�យ/ង;ន�

Page 52: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 51

( )( )( ) ( )( )( )( ) ( )( )

4 2 2 4 2 2 4 44

122 121 121 1

4

2

1

2

x x x y xy x x y

x y x y

x y y y

x y x y

y+ + + − − + =

=

+ − − − +

( ) ( ) ( ) ( ) ( )2 2 4 2 2 4 2 2 2 2414y x y yx x x y xy x x y xy y − −⇔ +− =

+ + + +

( ) ( ) ( )( )

( ) ( ) ( ) ( )

( ) ( )( )

( )

4 2 2 4 2 2

4 2 2 4

4 2 2 4 3 3 2 2

4 3 2 2 3 4

44 5

40

14 (7)

14

14 4 8

4 6 4

( ) 1 1

4 1

4 1

4 1

1

( )k k k

k

x y y y

x y y

x y y y

x x y xy x x y

x y x xy x y x y

x y x x

x y x

x y x y

xy x y

x y x y xy y

C x y x y−

=

⇔ + + −

⇔ + +

⇔ + + − −

⇔ − + − +

⇔ −

− − + =

− − +

= ⇔

+ =

− − =

− =

=− ⇔− =∑

=ង t x y= +

�6� ( ) ( )2 2 tyx x y x y+ −= =− (��Z� 1x y− = ),

( ) ( ) ( )2 2 2

2 2

2

1

2

x yy

x y tx

+ − =+ +

+ =

2 2 2 1

4 ( )21

121 122

( ) 1,

121( ) 1212

x yt

xy x y

ty x

t

x y y

y−= +

−− = − − =

=

− − − =

ព��6�ម� � (3)�P'� ( )4

51121

2 2

1243 3

t t tt t

−−⇔ = ⇔= =−

.*ច�ន� �យ/ង;ន 1x y− = ន�ង 3x y+ = �

Sញ;ន ច��5/យ�ប"�បព<នQគM� ( ; ) (2;1)x y = �

'()&'()&'()&'()&

Page 53: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១០

�. �����យ�បព<នQម� �� 4 3 2 2 3 4

3 4

2 2 12 8 1 0 (1)

2 1 (2)

x y x y xy yx

x y y

+ − − + + =

+ =

�. �កU'ច�ន�នគ�"�ប"ម� �� 3 4 33 10 16 0xx y+ + − =

1. �គ !���� : ABC � =ង , ,M N P ��@ងA� 4���2�5/ប,- �ជmង , ,BC AC AB ,

ប,- ច�ន�ច 1 1 1, ,A B C ��@ងA� 'ច�ន�ចក,- 5�ប" , ,AM BN CP �

បfg ញJផ5�ធ@ប�ក�Cផ7���� : 1 1 1A B C ន�ង���� : MNP ម�នb�<យនDង

ទ�=�ង�ប"ប,- ច�ន�ច , ,M N P �

E. �គ ! , ,a b c 0> ន�ង 2 2 2 12ba c+ + = �

�ក��C5ធ�ប�ផ���ប"ក�នyម� 3 32 2 23 2 2 2. . .c c aS a b b c a b= ++ + + +

K. ��យបI% ក"J ព� 2011ច�ន�នគ�"# �ជ%&ន,កL��យ �គ)�bច��ជ/យក;នព��

ច�ន�ន).5ផ5ប*ក � Mផ5.ក�ប"ព�ក?)ចក�ច"នDង 4018 �

N. ក��ងប3ង" Oxy , �គ !���� : ABC &នក*F����នប,- ក�ព*5គM ( )40; , 1;0 ,

3BA

( )1;0C � =ង ( )P '�ន���គប"ប,- ច�ន�ច M ក��ងប3ង").5&ន�ប)#ង�P BC �n/នDង

មធ!មធ�:� &��Cន�ប)#ង�P AB ន�ង AC � =ង I 'ផa���ងBង"\� Dកក��ង�ប"���� :

ABC � �ក�គប"ប,- �មគ�:ម���ប"ប,- ប67 �" �"=ម I �O/យ&នច�ន�ចប���មA�

នDង ( )P � '()&'()&'()&'()&

ចំេល�យ

�. ជ�ន� (2)ច*5 (1) �យ/ង;ន�

4 3 2 2 3 44 2 12 9 0x y x y xy yx + − − + = .

4 2 2 4 2 2 3 34 9 6 12 4 0x y y x y xx xy y⇔ + + − − + = .

( )22 2 2 22 3 0 2 3 0 (*)x xxy y xy y⇔ + − = ⇔ + − =

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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��យ 0y = ម�ន�ផ7GងH7 �" (2) �6��យ/ងព�ន��! 0y ≠ �

�ព5�6�, 2 1

2(*)3

. 3 0

3

x

x yyx x

x x yy y

y

= = − = = − = −

⇔ ⇔ ⇔

+

i �ប/ x y= �6�ព� (2) �យ/ង;ន� 4

4

1 1

3 3y y ±== ⇔

i �ប/ 3x y= − �6�ព� (2)�យ/ង;ន� 4 153y− = (ម�នម�O��ផ5)

.*ច�ន� �បព<នQ).5 !&នច��5/យព�� 4 4 4 4

1 1 1 1; , ;

3 3 3 3

− −

�. ព�ន��!ម� �� 3 4 33 10 16 0 :xx y+ + − =

∗ �ប/ 0 (mod 3) 1(mod 3)x y≡ ⇒ ≡

ម� �).5 !�P'� 2 4 33(3 ) 10(3 1)( ) 16 03 m nm + + + − =

9 15k⇔ = , ម� �ម�ន&នU'ច�ន�នគ�"�

∗ �ប/ 1 3) (mod 3( od )m 0yx ≡ ⇒ ≡

ម� �).5 !�P'� 3 4 33(3 1) 10(3 ) 0( 163 1) m nm + + + − =+

9 12:k⇔ = ម� �ម�ន&នU'ច�ន�នគ�"

∗ �ប/ 1(mod 3) 1( 3)y ox m d≡ − ⇒ ≡ −

ម� �).5 ! 3 យ�P'� 3 4 33(3 1) 1( 0(3 1) 163 1) 0m m n+ − + − − =−

9 15:k⇔ = ម� �ម�ន&នU'ច�ន�នគ�"�

.*ច�ន� ម� �).5 !ម�ន&នU'ច�ន�នគ�"�ទ�

1.

N

B

A

C

1A

1B

1C

M

P

'C

'A

2B

2A

2C

'B

G

Page 55: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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=ង ', ', 'A B C ��@ងA� 'ច�ន�ចក,- 5�ប" , ,BC CA AB , ប,- ច�ន�ច 2 2 2, ,A B C =ម

5��ប"ឆ3��A� នDង 1 1 1, ,A B C �ធ@បនDងប,- ច�ន�ចក,- 5�ប" ' ', ' ', ' 'B C C A A B �O/យ

G 'ទ��បជ��ទ�ងន"�ប"���� : ABC � �ព5�6� =មចtប"ប�)5ង\�ងផa�� G =ម

ផ5�ធ@ប 1

2− ប�)5ង���� : ABC �P'���� : ' ' 'A B C �

�យ/ង;ន� 2 12 2

2 1

A A MBk

B CA B

A A MA C

C Ck

B

′ ′ ′= ′⇒ =′

=′

= −� �

�O/យ MB kMC= −� �

2A⇒ )ចក ' 'B C =មផ5�ធ@ប k− �O/យ M )ចក BC =មផ5�ធ@ប k−

)�=មចtប"ប�)5ង\�ងផa�� G =មផ5�ធ@ប 1

2− ប�)5ង BC �P' ' 'B C

⇒ ចtប"ប�)5ង\�ងផa�� G =មផ5�ធ@ប 1

2− ប�)5ង M �P' 2A

.*ចA� ).�� ចtប"ប�)5ង\�ងផa�� G =មផ5�ធ@ប 1

2− ប�)5ង N �P' 2B , ប�)5ង P

�P' 2C �

⇒ ចtប"ប�)5ង\�ងផa�� G =មផ5�ធ@ប 1

2− ប�)5ង MNP∆ �P' 2 2 2A B C∆

2 2 2

1

2A B C MNPS S⇒ =

�យ/ង��e# ���យបI% ក"J� 1 1 1 2 2 2A B C A B CS S=

=ង 1 1' ' , , , ,C A b A B c B AB C Ba x C y′ ′ ′ ′ ′ ′= = = == ន�ង 1'A C z= ( , , , , , 0)a b c x y z > �

�យ/ង;ន� 1 1 1 1 1' ( )

' '. ' '

.A B C

A B C

AS A B z a x

S A B A C

C

bc′ ′ ′

−′= =

.*ចA� ).�� 1 1 1 1( ) ( )

;B A C C A B

A B C A B C

S Sx c z y a x

S ac S ab′ ′

′ ′ ′ ′ ′ ′

− −= =

1 1 1 1 1 1

.A B C B A C C A B A B C

abz bcx acy ayz bxz cxyS S

abcS S′ ′ ′ ′ ′ ′

+ + −+ + = − −⇒

��យបI% ក".*ចA� ).� �យ/ង;ន�

2 2 2 2 2 2

.A B C B A C C A B A B C

abz bcx acy ayz bxz cxyS S

abcS S′ ′ ′ ′ ′ ′

+ + − −+ + = −

1 1 1 2 2 2A B C A B CSS⇒ = .*ច�6� 1 1 1

1

4A B C

MNP

S

S=

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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.*ច�ន� ផ5�ធ@ប�ក�Cផ7���� : 1 1 1A B C ន�ង���� : MNP ម�នb�<យនDងទ�=�ង

�ប"ប,- ច�ន�ច , ,M N P �ទ�

E. =ង 3 3 32 2 2 2 2 21 2 3, ,a b c S b c a S aS c b= + = + = +

�យ/ង;ន� ( ) ( ) ( )3 22 2 2 2 2 2631

1.

1. .8

22

28S b a aa c b= + = +

Fន�#�-នj# �មiពក*��ច��Z� 6 ច�ន�ន�6�គM 2 2 2 2 2 2 2, 2 , 2 ,2 , , 8a ba a c b c+ +

�យ/ង;ន� ( ) ( )2 2 2 2 2 2 2

1

2 81

2 6

22a b cbS

a a c+ + + ++ ++≤

( )2 2 2

1

6

12

2 8a b cS

+ ++≤

.*ចA� ).�� ( ) ( )2 2 2 2 2 2

2 2

6 8 8,

12 1

2

2

2b c cS

a bS

a+ + + + +≤ ≤

+

.*ច�ន� �យ/ង;ន� ( )2 2 2

1 2 3

1012

12

a b cSS SS

+ ++ + =≤=

K. �ព5)ចកច�ន�នគ�"# �ជ%&ន,ម�យនDង 4018�6�ប,- ច�ន�ន�:5"��e#4���2ក��ង

�ន�� { }...,0, 4017 �

ក��ងប,- �:5"�ង�5/ �យ/ង)ចក�ចញ'�កmមន�ម�យ.*ច�ង�� ម�

+ �កmមទ�ម�យ ��ម&នប,- ច�ន�ន�ព5)ចកនDង 4018;ន�:5"�n/ 0

+ �កmមទ�ព����ម&នប,- ច�ន�ន�ព5)ចកនDង 4018;ន�:5"�n/ 1� M�n/ 4017

+ �កmមទ�ប���ម&នប,- ច�ន�ន�ព5)ចកនDង 4018;ន�:5"�n/ 2 � M�n/ 4016

.............

+ �កmមទ� 2009 ��ម&នប,- ច�ន�ន�ព5)ចកនDង 4018;ន�:5"�n/ 2008 � M�n/ 2010

+ �កmមទ� 2010 ��ម&នប,- ច�ន�ន�ព5)ចកនDង 4018 ;ន�:5"�n/ 2009�

.*ច�ន� &នS�ងF" 2010�កmម, )�)ប�'&ន 2011ច�ន�ន �6�=ម�ទD-�បទ Dirichlet

�?ងព�ក? ��e#&នព��ច�ន�ន ).5�:5"ក��ង�ប&:#�ធ�)ចកនDង 4018 o3 ក"ច*5ក��ង�កmម

'ម�យA� �

�ន� គM'ព��ច�ន�ន).5��e#�ក ��Z��ប/ព��ច�ន�ន�ន� &ន�:5"�n/A� �6�ផ5ង�ប" ព�ក?)ចក�ច"នDង 4018, �ប/ព�ក?&ន�:5"�ផpងA� �6�ផ5ប*ក�ប"ព�ក?)ចក�ច"

Page 57: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 56

នDង 4018 �

N. ម� �ប67 �" ,AB AC ន�ង BC ��@ងA� !��យ�

4 4( 1), 0( 1),

3 3y yx xy = −+ − == �

ច��Z�ច�ន�ច 0 0)( ;M x y ,កL��យ �6��ប)#ងព� M �Pប,- ប67 �"�ង�5/=ម5��ប"គM�

1 2 3 0

4 3 4 4 3 4, ,

5 5

x y xy

yd dd= =

+ −=

− +

.*ច�6� 2 0 0 0 023 1 2 0

4 4 4

5

4 3. .

3

5

y yd

x xd yd

− + + −= ⇔ =

2 20 0 0

2 20 0 0

2 3 2 0(*)

8 1 1

2

7 2 8 0

y y

x y y

x + + − =⇔

+ − =

Sញ;ន �ន��ច�ន�ច ( )P ��ម&នព��)ផ�កគM�

2 2 2 21 2) : 2 2 3 2 0, ( ) :8 17 12 8 0( x y y P x y yP + + − = − + − = .

�ឃ/ញJ ( 1;0), (1;0)B C− កL'ព��ច�ន�ច��មA� �ប" 1 2),( ( )PP �

ចt",", ផa���ងBង"\� Dកក��ង I �ផ7GងH7 �" 1 2 2d d d= = ,

�6��យ/ងគ:6;ន 1

1(0; )

2( )I P∈ �

ព�ន��!ប,- ប67 �" �"=ម I &ន^ង 1) ,:(

2y kx kd = + ∈ℝ� �យ/ង;នប,- ក�:� �

+ �ប/ 0k = �6� 1

2y = , ជ�ន�ច*5�បព<នQ (*) , �យ/ង�ឃ/ញJ ( )d �ផ7GងH7 �"5កqខ:r

�បoន�

+ �ប/ ( )d �"=ម ( 1;0), (1;0)B C− �6� 1

2k = ± , &នប67 �"ព��).5��e#A� គM (2 1)x y±= −

�O/យព�ន��!S�ងព��ក�:� �ន� �ទQ)��ផ7GងH7 �"5�l�"�

+ �ប/ 1

20,kk ≠ ≠ ± �6�ជ�ន�ច*5��យH7 5" 1

2y kx= + ច*5ម� ��ប" 1 2),( ( )PP �

ម� �bប"��ច�ន�ច�បពB�ប" ( )d ន�ង 1( )P គM�

2 21 12 )2( 3( ) 2 0

2 2kx kxx + + + − =+

2 2) 5 ] 0[(2 0xx k k x+ + = ⇔ =⇔ � M 2

5

2 2

kx

k= −

+

Page 58: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 57

��យ��C5S�ងព���ន��ផpងA� �6� ( )d ��e# �" 2( )P ��ង"ច�ន�ច)�ម�យគ�" � Mម� ��

2 21 18 )17( 12( ) 8 0

2 2kx kxx − + + − =+

2 2 25( 17 ) 58 0

4x kxk⇔ − −− = &នU)�ម�យគ�"

2

2 2 2

2 348 17

1725

( 5 ) 0

0

8 17 0, 4(

2

8 217 )4

k k

kk

kk

=− − = =

±=⇔ ⇔

− ≠ +

−±

.*ច�ន� &នS�ងF" 7 ��C5�ប" k �ផ7GងH7 �"��:/ ��បoនគM�

1 2 34 20, , ,

2 17 2k kk k= ± = ± = ±= �

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១១

�. �����យ�បព<នQម� ��

2

2

2

( ) 3

( ) 8

( ) 1

y z

y z x

z x

x

y

− =

− = −

− = −

�. �����យម� �� 4 4 4 41 2 3 13... 30x xx x+ + + = ( ix ∈ℕច��Z� 2, ..1 13, .,i = )

1. �គ !���� : ABC � =ង ( )), (( ),n pm ��@ងA� 'ប,- ប67 �" �"=ម , ,A B C ន�ង

)ចកប� �&������ :'ព��� បfg ញJ ( ), ( ), ( )m n p �បពBA� ��ង"ច�ន�ចម�យ�

E. �គ !���� : 1 1 1A B C ន�ង 2 2 2A B C ).5&ន�ក�Cផ7 1 2,S S ន�ងប,- �ជmង��@ងA�

គM 1 1 1, ,a b c ន�ង 2 2 2, ,a b c � បfg ញJ�

( ) ( ) ( )2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 1 2 2 2 1 2 2 2 1 216 .b a c ba c b a ac c S Sb+ − ++ − + − ≥+

K. ក��ង ��&ន�ជmង�n/ 4cm ,�គ�ក" 2011�ងBង").5&នFង̀�"ផa���n/ 1

30cm ច*5�Pក��ង

���6�� ��យបI% ក"J &នប67 �"ម�យ�បពBនDង 17�ងBង"�ង�5/�

Page 59: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 58

N. ក��ងប3ង" Oxy �គ !;9 9̂ប*5 2 2

( ) : 304 4

x yH − = � ��យ.DងJ ���� : ABC &ន

ក�ព*5S�ងប�4���2�5/ ( )H � ��យបI% ក"J F��*ង" K �ប"���� : ABC កL

4���2�5/ ( )H ).�� '()&'()&'()&'()&

ចំេល�យ

�. �យ/ង&ន� 2

2

2

( ) 3 (1)

( ) 8 (2)

( ) 1 (3)

y z

y z x

z

x

x y

− =

− = −− = −

ប*ក (1), (2), (3)�យ/ង;ន� 2 2 2( ) ( ) ( ) 6y z y z x z xx y− + − + − = −

គ�: (1), (2), (3) �យ/ង;ន� 2 2 2.( )( )( ) 24y z y z z x xx y− − − =

� M 2 2 2 2 2 2.(6) 24 2. .4y z y zx x x y z⇔ = ⇔ == ±

�ង̀��ឃ/ញJ� . . 0x y z ≠

• ក�:� ទ��� *. (. )2x y z =

គ�: (1)ន�ង y �O/យគ�: (2)ន�ង x ប*កប�a* 5A� �យ/ង;ន�

. . ( ) 3 8x y z y x y x− = − � M 1

6x y=

គ�: (2) ន�ង z �O/យគ�: (3)ន�ង y ប*កប�a* 5A� �យ/ង;ន�

. . ( ) 8x y z z y z y− = − − � M 1

10z y=

ជ�ន�នច*5 3 31 1(*) . 2 2

6 10120 15y y y y y⇔ == ⇔=

3 3

3

15 5

5

1x z⇒ ⇒= = �

• ក�:� ទ��� . . 2x y z = − (**)

គ�: (1)ន�ង y �O/យគ�: (2)ន�ង x ប*កប�a* 5A� �យ/ង;ន�

. . ( ) 3 8x y z y x y x− = − � M 1

2x y=

Page 60: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 59

គ�: (2)ន�ង z �O/យគ�: (3) ន�ង y ប*កប�a* 5A� �យ/ង;ន�

. . ( ) 8x y z z y z y− = − − � M 1

2z y= −

ជ�ន�ច*5 (**) 31 1. .( ) 2 2 1

2 28 1y y y y xy z⇔ = ⇔ ⇒ == − ⇒− = = − �

�. ច��Z� n គ*� 4 216 0 (mod16)n k= ≡

ច��Z� n �� 4 21 ( 1)( 1)( 1) ( )mo0 16dn nn n− = + − + ≡

�6� 4 1(mod16)n ≡

.*ច�6� 4 4 4 41 2 3 13... 1(m 6)odx x px x+ + + + ≡ , ច��Z� { }1, 2, 3, ...,10, 3 (*)p ∈

)���យ 14 (mod16)30 (**)≡

ព� (*) ន�ង (**) : ម� �).5 !An នU�

1. =ង ', ', 'A B C ��@ងA� 'ច�ន�ចក,- 5�ប" , ,BC AC AB � ព�ន��!�ម/5ច�ន�ច D 4���2

�5/ ' 'B C 89ង, ! 'A D )ចកប� �&������ : ' ' 'A B C 'ព��)ផ�ក�n/A� , 'p 'កន3�

ប� �&��, ', ', 'a b c 'ប,- �ជmង���� : ' ' 'A B C

' ' ' ' ' ' 'DC A C p DC p b⇒+ = = − , .*ចA� ).� ' ' 'DB p c= −

.*ច�6� ( ' ') ( )

'

( ) ( )(1)

A C p c A B ACp b p b

a a

p c ABA D

− − −′ ′ ′ ′ ′ ′+ − − −′ = =� � � �

=ង I 'ផa���ងBង"\� Dកក��ង���� : :ABC

.. . 0IA b Ia B c IC+ + =� � � �

) .( ) .( ( ) 0. A A A I b A B Aa I c A C A I′ ′ ′ ′ ′ ′⇒ − + − + − =� � � � � � �

( ) .1 1

2 2.2 A I a AB AC b BC c Cp B′⇔ = − + − +

� � � � �

) .1 1 1

( ( (2 2

) . )2

AB AC b AC A Ba B c AC A+ − − + −= −� � � � � �

) (( )ABp p b ACc= −− − −� �

(2)

ព� (1) ន�ង (2)�យ/ង;ន ', ,A I D ��"��ង"ជ��

.*ចA� ).� ប,- ប67 �" �"=ម ', 'B C )ចកប� �&������ : ' ' 'A B C 'ព��)ផ�ក�n/A�

�O/យ�បពBA� ��ង" I , ប,- ប67 �"�ន�'�*បiព�ប" ( )), (( ),n pm =មចtប"ប�)5ង\�ង

Page 61: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 60

ផa�� G =មផ5�ធ@ប 1

2k = − ច��Z� G 'ទ��បជ��ទ�ងន"���� : ABC

.*ច�ន� ( ), ( ), ( )m n p �បពBA� ��ង" 'I '�*បiព�ប" I =មចtប"ប�)5ង\�ងផa�� G =ម

ផ5�ធ@ប ' 2k = − �

E. �យ/ង&ន ��ង̀�.*ច�ង�� ម� ច��Z� 0 α π≤ < �6�

2 2 2 22 .cos ( .cos ) .sin 0y xy x y yx α α α+ − = − + ≥

Iw �n/�ក/�&ន x y⇔ = ន�ង 0α = �

Fន�#�-នj ��ង̀��ង�5/ ច��Z� 1 2 2 1,cx y b cb == ន�ង 1 2A Aα = −

ន�ង 2 2 2 2 2 21 1 1 1 1 1 2 2 2 2 2 22 ,.cos 2 .cosc A b c a b c A bb c a= + − = + −

�6��យ/ង;ន� 2 21 2 2 1 1 2 2 1 1 2( ) ( ) 2 cos( ) 0c b c b c c Ab b A+ − − ≥

2 21 2 2 1 1 2 2 1 1 2 1 2 2 1 2( cos) ( cos) 2 si i2 n s n 0c b c b c b c b cb A AbA Ac⇔ + − − ≥

2 2 2 2 2 2 2 21 2 2 1 1 1 1 2 2 2 1 2) ( ) ).( ) 8 . 0

1( (

2b c b c c a b c a Sb S⇔ + − + − + − − ≥

ព63 � ��ច���5�យ/ង;ន�

2 2 2 2 2 2 2 2 2 2 2 21 2 2 2 1 2 2 2 1 2 2 2 1 2( ) ( ) ( ) 16 .c b a b a c ba c b a c S S+ − + + − + + − ≥ �

K. ង" 124 ប67 �"�បA� �PនDង�ជmង ��ម�យ, �3 �A� 4

125cm ព�ក?)ចក ��' 125ច���@ក

ច��� :)កង&នទទDង 4

125cm

��យ 1 4

30 125> �6��ងBងន�ម�យ��ទQ)���e# �"

89ង��ច ��យប67 �"�ង�5/89ង��ចម�យ..

&ន 124 ប67 �", 2011�ងBង" �O/យ 2011 )ចក 124

;ន 16 �:5" 27 �6�=ម�ទD-�បទ Dirichlet

&នប67 �"ម�យ �"89ង��ច 17�ងBង"ក��ងច��,ម 2011�ងBង"�ង�5/�

N. ព�ន��!�ម/5 Lemma �ង�� ម�

! ( )H ')ខp� ង&នម� �� my

x= � �ប/���� : ABC &នក�ព*5ប�4���2�5/ ( )H

�6�F��*ង" K �ប"?កL4���2�5/ ( )H ).��

Page 62: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 61

��យបI% ក" :Lemma

( ,( , ), ) ),, (m m

B b C cm

A aa b c

, 4���2�5/ ( )H ច��Z� ,,a b c ខ�A� ព�ម�យ�Pម�យន�ងខ�ព� 0

�យ/ង;ន ( , )K u v

),( ), ( ,AK u a v BC cm m m

bb

a c= − − = −−

� �

),( ), ( ,BK u b v AC cm m m

aa

b c= − − = −−

� �

K 'F��*ង"���� : AK BC

BA

K ACBC

⊥⇔

� �

� �

( ).( ) ( ).( ) 0

( ).( ) ( ).( ) 0

m m mu a c b v

a c bm m m

u b c a vb c a

− − + − − = − − + − − =

( ). 0

( ). 0

m mu a v

a bcm m

u b vb ca

− − − = − − − =

2

2

0( )

0

m mu a v

mbc abc v b a a babcm m

u b vac abc

− − + = − = −

− − + =⇔

2

.abc m

v u v mm ab

uc

K⇒ ⇒ == − =⇒ ⇒− 4���2�5/ ( )H

�យ/ង;ន 2 2

( ) 1120 120

:x y

H − =

Fន�#�-នjចtប"ប�)5ងF<កp=ម�*បមន-�ង�� ម� ' '

' '

x x y

y x y

= + = −

�យ/ងប�)5ង ( )H �P'^ង� my

x= ច��Z� 30m = �យ/ង;នបIg ��e#��យបI% ក"�

'()&

Page 63: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១២ �. zប&J x 'U'ច�ន�នព��ខ�ព�*ន!�ប"ម� � 2 0bx cax + + = ,ច��Z�

, ,a b c 'ប,- ច�ន�នគ�"�ផ7GងH7 �" | | | | | | 1a b c+ + > �

��យបI% ក"J� 1| |

| | | | | | 1x

a b c+ + −≥ �

�. �កច�ន�នគ�"# �ជ%&ន n ).5�ផ7GងH7 �"�

( )1 4 ( 1) 42 2 2 2 2 21 1 3 2 3 8 693n n n n n nn n n nn nn n n + + + + − − − = − + + + + + +

+ + +

�2ទ��ន�, [ ]x 'ច�ន�នគ�"ធ�ប�ផ��ម�ន�5/ព� x �

1. �គ !ច�ន�ច M 4���2)ផ�ក�ងក��ង���� : ABC � =ង 1 2 3, ,d d d ��@ងA� '�ប)#ង

ព� M �P�ជmងS�ងប� , ,BC CA AB �

��យបI% ក"J� �ប/ 31 2 3. .dd d r≥ �6��យ/ង;ន OM OI≤ , ក��ង�6� ,O R'ផa��ន�ង

��ងBង"\� Dក��], ,I r 'ផa��ន�ង ��ងBង"\� Dកក��ង���� : ABC �

E. �គ ! , ,x y z 'ប,- ច�ន�នព���ផ7GងH7 �"� 10 36xyz ≥ + �

��យបI% ក"J� 3 2 3 2 3 2

1

2

y z x

x y y zz x yz x≤

+ + ++ +

+ + + �

K. បfg ញJ ច��Z��គប"ច�ន�នគ�"ធមn'�� 2n ≥ &ន�ន�� S ��ម&ន n ច�ន�នគ�"ធមn'��

89ង, ! ab )ចក�ច"នDង 2( )a b− ច��Z��គប" a b≠ �ផpងA� �O/យ4���2ក��ង S �

N. �គ !���� : ABC '���� :)កង ).5�ជmងF��ប9*��ន�គM AB � =ង ,a bm m ��@ង

A� 'ប67 �"�ម.uន�ប"�ជmង ,BC AC �

��យបI% ក"J� 5 3

2 2a bmm

a b

++

≤ ≤ �

'()&'()&'()&'()&

Page 64: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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ចំេល�យ

�. ��យ ,,a b c 'ប,- ច�ន�នគ�"�6� 2a b c+ + ≥ �

បIg ��e#��យបI% ក"��e#;ន����k/ង# �ញ� ( ) 1 (1)a b c x x+ + > +

�ប/ 1x ≥ �6� ( ) 12a b c x x x≥+ + > +

�ប/ 0 1x< < �6� 2x x>

.*ច�ន� �យ/ង;ន 2 2ax bx ca x b x a x cb x≥ ≥ + = −+ =+

�ព5�6� ( ) ( )1a b c x c x+ + > +

.*ច�ន� (1) ព��ច��Z� 0c ≠ � M 1c ≥ � �./ម0�ប�a ប"5�l�" �យ/ងព�ន��!�ព5 0c = ,

�ព5�6� 0ax b+ = �

��យ 0x ≠ �O/យ 2a b c+ + ≥ �6� 0a ≠ �O/យ 0b ≠ � M 1a ≥ ន�ង 1b ≥

ព��6� FងR�ង�ឆBង�ប" (1) ��e#;ន����k/ង# �ញ�

( ) ( )1 1a b x ax b x b b x b x x+ = + = − + = + ≥ + �

.*ច�ន� 5�l�"��e#;ន��យបI% ក"��ច^5"�

�. �យ/ងនDង��យបI% ក" ��� �# �មiពS�ងព���ង�� ម�

2 2 23 1 1 (12 4 )2 1n n n n nn n< < + <+ ++ − + +

2 2 23 2 8 3 (2 )1 2 24 2n nn nn n n n+ < + < <+ ++ + + +

�យ/ង��យបI% ក" (1)

2 2 32 4 4n n n= < +

ន�ង 2 2 2 21 1 2 11 2n n nn n nn n− + + + + ++ < + = +

ក��ង (1), �យ/ង��e# ���យបI% ក"J

2 2 23 1 14 nn n n n< ++ − + + + .

2 2 2 12 1 2 1n n nn n⇔ + < − + + +

4 2 4 24 1 4 44 4nn n n⇔ + + < + + ក�:� �ន� ព��'ន�ចaច��Z��គប" n �

�k*# �យ/ង��យបI% ក" (2), �យ/ង&ន�

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Page 64

2 2 2 22 1 3 22 1 n n nn n n n n+ + + + += < + ++ .

ន�ង 2 24 48 3 8 4 2 2n n nn n+ + + <+< +

ក��ង (2) , �យ/ង��e# ���យបI% ក"�

2 2 23 842 3n n nn nn+ + + +< ++

2 2 23 2 42 2 1n nn n nn⇔ + + + + +<

4 3 2 4 3 216 20 8 44 16 20 8 1n n n n n nn n⇔ + + + < + + + +

ក�:� �ន� ព��'ន�ចa �6� (2) ��e#;ន��យបI% ក"��ច^5"�

�ព5�6�, ព�ម� ��យ/ង;ន�

2 ( 1)(2 1)( 1)2 (2 1) 2 69n n n n n n n+ + + − − − + = .

4 1 69 17n n⇔ ⇔+ = = .

.*ច�ន� Uម� �).5 !គM 17n = �

1. =ង 1 1 1, ,A B C 'ច��,5)កង�ប" M ��@ងA� �P�5/ , ,BC CA AB �

�យ/ង=ង ABCS ន�ង 1 1 1A B CS '�ក�Cផ7�ប" ABC∆ ន�ង 1 1 1A B C∆

�យ/ង;ន� � � �0 0 01 1 1 1 1 1

ˆ ˆ ˆ180 , 180 , 180A MB C B MC A C MA B= − = − = −

�6� 1 1 1 1 2 2 2 3 12 sin. . sin . sinA B CS C dd d d A d d B+= +

1 2 31 2 3

sin sin sinA B Cd

d d dd d

= + +

=ម�ទD-�បទFន�គមនj��ន��6� sinsin , , sin2 2 2

a b cA

R RB C

R== =

�6� 1 1 1

1 2 3

1 2 3

(12 )2A B C

d a b cS

R d

d

d d

d + +

=

មu9ង�ទ@� �យ/ងកL&ន� 1 2 3 (22 )ABC ad bS d cd= + +

ព� (1)ន�ង (2) �យ/ង;ន�

( )1 1 1

1 2 31 2 3

1 2 3

4 .2A B C ABC

d d d a b cS ad bd cd

R dS

d d

= + + + +

=ម# �មiព Bunyakovski �យ/ង;ន�

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1 1 1

21 2 3. (3( )2

)4 A B C ABC

ddS a b c

R

dS +≥ +

Fន�#�-នj�ទD-�បទ Euler ក��ង���� : 1 1 1A B C �យ/ង;ន�

( )1 1 1

2 2

24A B C ABC

OMRS S

R

−=

Sញ;ន� 1 1 1

2 2

2.4

44)4 (A B C ABC ABC

OMRS S

RS

=

ព� (3) ន�ង (4) Sញ;ន� 3 2

2 22

.2

( )4

( )

ABC

a b cRR r

SOM

rR

+ + =− ≥ � M 2 22RrR OM− ≥

=មទ�6ក"ទ�នង Euler គM� 2 22RrR OI− = ( I 'ផa���ងBង"\� Dកក��ង)

.*ច�ន� �យ/ង;ន� OM OI≤ (បIg ��e#��យបI% ក") �

E. ព�# �មiព Cauchy Schwarz− �យ/ង&ន�

2 4 2 2 2

3 21

( )

1

( )

1 1

x y z x y z y x y zx y

x y x y xy yz −+ = ≥ + + + ++ +

+ ++ =

+ +

.*ច�6� 3 2 2

1

( )

y xy y

x y x y zz

+ ++ +

≤++

.*ចA� ).�, �យ/ង;ន

3 2 3 2 3 2

y z x

x y y zz zx yx+ ++

+ + ++

+ 2 2 2

1 1 1

( ) ( ) ( )

xy y yz z zx x

x y z x y z x y z

+ + + + + ++ ++ + + + + +

2

3

( )

xy yz zx x y z

x y z

+ + + + + +=+ +

�./ម0�ប�a ប", �យ/ង��e# ���យបI% ក" 2

3 1

(( )

) 21

xy yz zx x y z

x y z≤+ + + + + +

+ +

ព��'.*ច�ន�, 2 2( ) 2( ) 6(1) ( ) xy yz zx x y zx y z⇔ ≥ + + + + ++ ++

2 2 2 2( ) 6y z x y zx⇔ + + ≥ + + + .

មu9ង�ទ@��យ/ង&ន ( )2 2 2 2( )3 y z x yx z⇔ + + ≥ + + , �6��យ/ង��e# ���យបI% ក"J

2 6(( 18) )xx y z y z≥ + + ++ + .

ក�:� �ន�ព��ច��Z� 3( 3)1x y z+ ≥ ++ ��Z�=ម# �មiព Cauchy �យ/ង&ន

333 3 10 6 3 3(1 3)x y xz yz≥ ≥ + = ++ +

# �មiព��e#;ន��យបI% ក"��ច^5"� Iw �n/ �ក/�&ន5���=)��

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 66

3 2

1 1

1 3

x y z

x y

x y z

= =

= = = +

, �បព<នQ�ន� An នច��5/យ, �6�Iw �n/ក��ង

# �មiព�ប"5�l�"ម�នbច�ក/�&ន�ទ�

K. �យ/ង��យបI% ក"5�l�" ��យ# �ធ�# �\�Fន�&ន��មគ:� �# �ទu�

ច��Z� 2n = យក { }2 ;10S = �

zប&J 5�l�"ព��.5" n k= &នន<យJ �យ/ងយក;ន�ន�� kS �ផ7GងH7 �"5�l�"�

�យ/ងនDង��យបI% ក"J 5�l�"ព��ច��Z� 1n k= + �

=ង L 'ពO�គ�:��ម�*ចប�ផ���ប"ប,- ច�ន�នខ�ព� 0&ន^ង 2( )a b− ន�ង ab ច��Z�

�គប"ប,- �បព<នQ , ka b S∈ � ព�ន��! { } { }1 | 0k kS L a Sa+ += ∈ ∪

Sញ;ន 1kS + &ន 1k + o��� �យ/ងនDង��យបI% ក"J? �ផ7GងH7 �"5�l�"�

ព��'.*ច�ន� �

�ប/ ម�យក��ងច��,មព��ច�ន�ន a � M b �n/ 0 �6� 2( )aab b−⋮ �

�ប/ ច�ន�នS�ងព��&ន^ង L a+ ន�ង L b+ �6��យ/ង;ន

( ) ( ) ( )L a L b l L a b ab ab+ + = + + + ⋮ .

( )( ) [ ]2( ) ( )L a L b L a L b+ + + − +⇒ ⋮ �

ព��6� Sញ;នបIg ��e#��យបI% ក"�

N. ង"����យក*F����ន 'ម�យក�ព*5 A 'គ�"����យ, AC 'F<កp Ox � �ព5�6� ក*F����ន

ប,- ក�ព*5គM (0;0), ( ; ( ;0)), CB bA b a ច��Z� ;2

aD b

'ច�ន�ចក,- 5�ប" BC �

ង"ច�ន�ច ( ; ), ( ;0)E a b F a− − − � �ព5�6� EAF∆ មម*5នDង ABC∆ �

x

y

a

b

A

B

C

D

E

F

G

/ 2a

/ 2b

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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'ង�ន��P�ទ@� ;2

bG a − −

'ច�ន�ចក,- 5�ប" EF �

�យ/ង;ន ,a bm mAD AG= =

ព�# �មiព���� :ក��ង ACD∆ , �យ/ង;ន� 2

aAD AC CD b< + = +

ព�# �មiព���� :ក��ង AFG∆ , �យ/ង;ន� 2

bAG AF FG a< + = +

.*ច�ន� 3( )

2a bmm a b< ++

.*ចA� ).�, Fន�#�-នj# �មiព���� :ក��ង ADG∆ �យ/ង;ន

2

2( )2

a bAD AG a bDG≥ = + + + +

.*ច�ន� 5( )

2a bmm a b≥ ++

.*ច�ន� 5�l�"��e#;ន��យបI% ក"��ច^5"�

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១៣

�. �����យម� �� 3 2 31 2 0x xx − − − + = �

�. �កU'ច�ន�នគ�"�ប"ម� �� 3 3 2 2 2 100y x y yx x− − − = + �

1. �គ !ច��� :)កង ABCD &ន 2 , 2BA a C aB = = � យកFង̀�" AB �ធB/'Fង̀�"ផa��,

ង"�2�ង��]ច��� :)កង ន*#កន3��ងBង"ម�យ� ច�ន�ច M 4���2�5/កន3��ងBង"�6�,

ប,- ប67 �" ,MD MC �" AB ��@ងA� ��ង" ,N L � បfg ញJ 2 2

21

AL

A

N

B

B =+ �

E. �គ !ប�ច�ន�ន# �ជ%&ន , ,x y z �ផ7GងH7 �" 33 5x y z+ + ≤ �

��យបI% ក"J� 4 4 44 43 625 15 5 81 4 45 5xy z yz x zx y xzy+ + ++ ≥+ �

K. �គ ! ( 3)n n ≥ ច�ន�ច�2�5/ប3ង", ក��ង�6�ម�ន&នប�ច�ន�ច,��"��ង"ជ��A� �ទ� i% ប"�គប"

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 68

ប,- ប67 �").5 �"=មព��ច�ន�ចក��ងប,- ច�ន�ច�6�� ��យ.DងJ ប,- ប67 �"�6�

�"A� ព�ម�យ�Pម�យ �O/យម�ន&នច�ន�ច, ��]ព�ច�ន�ច).5 !�2�5/ប67 �"ព��

ក��ងច��,មប67 �"�6� �"=ម�ទ� ច*�គ:6ច�ន�នច�ន�ច�បពB�ប"�ប"ប67 �")�

ព��ប9��,c �ក��ងច��,មប67 �"�6� �

N. �គ !���� : ABC ,ប67 �"ព��ក��ង ន�ងព����]�ប"ម�� C �"ប67 �" AB ��ង" L ន�ង M .

��យបI% ក"J �ប/ CL CM= �6� 2 2 24BCAC R+ = , ( R ' ��ងBង"\� Dក��]�ប" ���� : ABC ) �

'()&'()&'()&'()&

ចំេល�យ

�. 5កqខ:r � 3 2x ≥

�យ/ង;ន� 3 2 3 2 01 (1)x x x− − − + =

( ) ( )3 32 3 2 31 2 51 2 3 2x x x xx x⇔ − + = − ⇔ − − = −+ −−

( )

2 3

2 332 23

9 27

21 2 1 43

5

x xx

xxx

− −⇔−− + − +

+ − =+

( )( )

( )( )2

2 332 23

3 9

21 2 1

333 1

4 5

x xxx

xxx

x −+ − + = +

+ +⇔

−− + − +

( )

2

2 332 23

3

31

3

5

9(2)

21 2 1 4

x

x

x

x x

xx

+ +⇔−− +

= + + =

+ +

�ង̀�� 2

32

5

3 9(3)

2

x

x

x+ +−

>+

ព��'.*ច�ន� 2 33 9 2(( ) 23 5)xx x⇔ + − ++ >

2 33 1 2 2xxx⇔ + − > − .

2 2 33 1) 4( 2)( xx x⇔ + − > − (��Z� 3 2x ≥ )

4 3 22 7 6 9 0x xx x⇔ + + − + > .

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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2 2 2 2) ( 3) 5 0( x x xx⇔ + + − + > (ព��)

Sញ;ន (3) ព���

�យ/ង��e# ���យបI% ក" FងR�ង�ឆBង ( )2 32 23

3(2) 1 2 (4)

1 2 1 4x

x

x

+= + <− + − +

ព��'.*ច�ន� 32 2 23 1) 1(4) ( 2 1xx x⇔ − + − + >

=ង� 3 2 1, 0x tt −= > , �យ/ង��e# ���យបI% ក"� 2 32 1 1tt t+ + > +

2 2 3 4 3 22 1) 1 3 6 4 0(t tt t t t t⇔ + + > + ⇔ + + + > (ព��)

Sញ;ន (4) ព��, 6� ! FងR�ង�ឆBង 2< < FងR�ង�- �

.*ច�6� (2) An នU

.*ច�ន� ម� � (1) &នU)�ម�យគ�"គM 3x = �

�. =ង ,x y d d= + ∈ℤ� �ព5�6� ម� �).5 !មម*5នDង�

3 3 2 2( ) 2( ) 100 0( ) y y d y yy d yd − − + − − + − =+ .

2 2 3 2(3 4 ) 100 0(3 4) d d yy d dd⇔ + − + − − =− .

ច��Z��គប" d ∈ℤ �6� 3 4 0d − ≠

5កqខ:r �./ម0� !ម� � (1) &នUគM� 0∆ ≥

2 2 3 24 ) 4(3 4)( 1 03 0 ) 0( d d d dd⇔ − − − − − ≥ ,

4 34 123 00 1600 0d dd⇔ − − + ≤ ,

3 3400) 04

(3 4)(3

400d d d⇔ − ≤ ⇔ ≤ ≤−

��យ d ∈ℤ �6� {2,3,4,5,6,7}d ∈

ច��Z� 2d = , �យ/ង;ន 228∆ = ,Sញ;ន , 56y x= = � M 8, 6xy = − = −

ច��Z� 3d = , �យ/ង;ន 1865∆ = (�\5)

ច��Z� 4d = , �យ/ង;ន 2688∆ = (�\5)

ច��Z� 5d = , �យ/ង;ន 255∆ = , Sញ;ន , 50y x= = � M 5, 0y x= − =

ច��Z� 6d = , �យ/ង;ន 2576∆ = (�\5)

ច��Z� 7d = , �យ/ង;ន 969∆ = (�\5)

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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.*ច�ន� ប,- U'ច�ន�នគ�"�ប"ម� �គM� ( 6; 8), (5;0)(8;6 ,, 5) (0; )− − − �

1. =ង ,P Q ��@ងA� 'ច�ន�ច�បពB�ប" CDនDង MA ន�ង MB �

=ង ;D yP x CQ= = �

�យ/ង;ន� APD QBC∠ = ∠

(ម��&ន�ជmង)កង��@ងA� )

APD QBPD BC

AD QCC⇒ ∆ ∆ ⇒ =∼

222

2

x axy a

ya⇔ =⇔ =

2 2 2 2 2 2 2 2( 2 ) ( 2 ) 4 4 4 4PC xQD x a y a ax a y ay a+ = + + + + + + +=+

2 2 2 24 4 4 2 2 4y a ax ay xy x ax y+ + + + + − += 2 2 2( 2 2 )4) (2x y a x ya axy− += + + = + + (��Z� 22xy a= )

2 (1)PQ= .

Fន�#�-នj�ទD-�បទ Talet (=�5), �យ/ង;ន�

MN ML MA MB AL BN AB

MD MC MP MQ PC QD PQ= = = = = =

2 2 2 2 2 2 2

2 2 2 2 2 2

AL BN AB AL BN BN

QD

AL

PC QD PQ PC PQ

+ +⇒ = = = =

+ (=ម (1) )

2 2

2 2 22

1AL BN

AABAB

L BN+

⇒ = + ⇒ = (បIg ��e#��យបI% ក")�

E. 4 4 44 4 4 45 (13 625 15 5 1 5 )8xy z yz x zx y xyz+ ++ + + ≥

�យ/ង&ន� 4 4 44 4(1) 15 25 15 15 9 5

24 45

5 9xy z yz x yz y xyz⇔ + + + ≥+ +

4 4 41 4 1 1 425 9 54

23

5 9z x y

z x y⇔ + + + ≥+ +

2 2 22 2 2

4 4 49 25 5

93

25x y z

x y z⇔ + +++ + ≥

=ង 2 2 2 2 2 2(0;0), ), ), )

3 3( ; ( 3 ; ( 3 5 ;

5O

x x yA x B x y z

yy

x zC x+ + ++ + +

B A

C D

M

N

P Q

L

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Sញ;ន� 2 2 22 2 2

4 4 4, 9 , 25

9 25x OA y AB z BC

x y z+ += = =+

FងR�ង�ឆBង OA AB OCBC= + + ≥

( )2

2 2 2 23 5

3 5x y z

x y z

= + + + +

+

( ) ( )2

223 33

23

8 369 9

15(15 ) 9 15

15xyz xyz

xyxyz z

≥ + ≥ +

=ង ( )23 15xyzt =

�យ/ង;ន� 3

3

3 5

3 5

3 15x y z xyz

x y z ≤

+ + ≥

+ +

33 1 13 5 0xyz t⇒ ≥ ⇒ < ≤

Sញ;ន� 36 36 369 36 27 .36 272t t t t t

t t t+ + −≥= −

2.36 27 72 27 45t≥ − ≥ − = .

3639 5t

t⇒ ≥+ ⇒ FងR�ង�ឆBង 53≥ (បIg ��e#��យបI% ក")�

K. =ង S 'ច�ន�ច�បពB).5��e#�ក�

�ប/ 3n = �6� 3S =

ច��Z� 4n ≥ �6�ច�ន�ចន�ម�យ�).5;ន ! �ទQ)�&ន�2�5/ប67 �"ព�� �"=ម �O/យ

ម�ន)មន'ច�ន�ច).5��e#�ក�

=ង P 'ច�ន�នប67 �"ក�ព�ងព�ន��!, �យ/ង;ន�

2 ! ( 1)

2!( 2)! 2n

n n nP C

n

−= = =−

��យ ប67 �"ព��,កL��យ កL �"A� , �6�ច�ន�នច�ន�ច�បពBគM�

2 2( 1) 1 ( 1) ( 1) 1. 1 ( 1)(

2 2 2 22)

8p

p p n n n nC n n nn

− − − = = − = − − −

��យច�ន�ចន�ម�យ�).5;ន ! �ទQ)�&ន 1n − ប67 �" �"=ម �6�ច�ន�ន.ង).5��e#

(ប�^ប") (# �មiព Cauchy )

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 72

គ��គM� 21

( 1)( 2)

2n

n nC −

− −=

�O/យច�ន�ន.ងច�ន�ច �ប" n ច�ន�ច�6�គM 21nnC − �

��យ ��]ព�ប,- ច�ន�ច).5 ! ម�ន&នច�ន�ច,).5&ន�5/ព� 2ប67 �" �"=ម

�6� ច�ន�ចន�ម�យ��25"គM;ន)�ម-ង)�ប9��,c ��

.*ច�ន� ច�ន�នច�ន�ច�បពB).5��e#�កគM� 2 21

( 1)( 2)( 3)

8p n

n nnC

n nS C −

−= − −−= �

N. �ប/ CL CM= �6� CML∆ )កងម;���ង" C �

=ង O 'ច�ន�ចក,- 5�ប" ML � ��ជ/�� /����យក*F����ន.*ច�*ប�

=ង , ,Oa B C cO bA O= ==

�ព5�6� ( ;0), ( ;0), ((0;0) 0; ), ( ;0), ( ;0), A a B b C c MO c L c− (��Z�2

MLOC OM OL c= = = = )

=ម5កq:�ប67 �"ព��, �យ/ង;ន�

2 2 2 2 2

2 2 2 2 2

( )

( )

AL AC AL AC c a a

LB CB LB CB b c b

c

c

+⇔ ⇔ − =− +

= =

2 2 2 2 2 2( ) )( ( () )b c bc c a ca⇔ + = − +−

2 2 2 2 0ac a b cb ba⇔ + − − =

2( ( ) 0)a b c ab⇔ − =−

2 2

2 ( ;0)c c

c b Ba a

ab⇔ ⇒== ⇔

4

2 2 2 2 22

( )BC ccAa

ac

C +

+

= +

+

24 4 2 2 2 22

(1)2

c a c a

a

ca + + = =

+

=ង ( ; )I x y 'ផa���ងBង"\� Dក��]���� : ABC �

�យ/ង;ន� 2 2 2 2

2 222

2 2 2 2 2

( )( )

( )

yx aAI CI AI

cAI BI

x y cCI

BAI x a xIa

y y

+ = + −=

⇔ ⇔=

−=

= − −

+

=

+

B A

C

O M L

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 73

( )

2 22 2

2 22 22 2 4 2

2 22 2

222

y cax cy aax cy aaaax a a xx

aa

ccccc c

−−⇔ ⇔ ⇔ ++

= − = − = = == −

2 2

;2

aI

a

cc

+

Sញ;ន 22 2

2 2 2 244a

Ra

cIC AC BC

=

+ = +

= (បIg ��e#��យបI% ក")

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១៤ �. �����យម� �� 23 51 134x x x+= − −+

�. �ក�គប"ប,- U'ច�ន�នគ�"�ប"ម� �� 5 5 5 5 55 25 125 625 0y z tx u+ + + + =

1. �គ !ច��� :)កង ABCD &នប67 �" AB ន�ង CD �"A� ��ង" E � =ង F 'ច�ន�ច

ក,- 5�ប" ,BC T 'ច�ន�ច�បពB�ប" AC ន�ង BD � ��យបI% ក"J ច��� :

ABCD 'ច��� :Z� យ5���=)� , ,E T F 4���2�5/ប67 �")�ម�យ �

E. ��យបI% ក"# �មiព� 2 2 2

9

4( )( ) ( ) ( )

m n p

m n pn p p m m n+ +

++≥

++ +, ច��Z�

, ,m n p 'ប,- ច�ន�ន# �ជ%&ន�

K. �គ ! n 'ច�ន�នគ�"# �ជ%&ន, 3n ≥ � ��យបI% ក"J ព� 22

n + 'ច�ន�នគ�",ម�យ

)�ង&នព��ច�ន�ន).5&នផ5ប*ក � Mផ5.ក)ចក�ច"នDង n �

N. �គ !���� : ABC ម;���ង" A� =ង M ច�ន�ចក,- 5�ប" ,AB G 'ទ��បជ��

ទ�ងន"���� : ACM � =ង I 'ផa���ងBង"\� Dក��]���� : ABC

បfg ញJ GI CM⊥ � '()&'()&'()&'()&

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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ចំេល�យ

�. 5កqខ:r � (*)1

3x ≥ −

�យ/ង;ន� ( )2 213 5 83 1 4 3 1 4 54 1x x x x xx x++ = − + = − +− ⇔ − −+

2 5) 13 1 (2 2x x x−⇔ − += −+

=ង 3 1 2 2x y+ = − , 5កqខ:r � 1 (**)y ≥

�ព5�6�, ម� ���e#;នប�)5ង'�បព<នQម� ��

2

2 2 3 1

2 2 (2 2) 5 1

y x

y xx

− = +

− = −− +−

2

2

3 1 (1)

(2 2) 5

(2 2)

2 1 (2)

x

x y

y

x

= +⇔

= −−

+

(2 2 2 2)(2 2 2 2) 2 2y x y x x y− + − − − + = − +⇒

4( 2)( ) 2( ) 0x y y x y x+ − − − − =⇔

2( )(2 2 5) 02 5 2

y xy x x y

y x

=− + − =

= −⇔ ⇔

ជ�ន� y x= ច*5 (1) , �យ/ង;ន�

2 23 1 11 3 0

11 73

8(2 2) 411 73

8

xx

x x

x

x

+=− = + ⇔ − + = ⇔ −=

ជ�ន� 2 5 2y x= − ច*5 (1) �យ/ង;ន�

2 23 1 15 8 0

15 97

8(3 2 ) 415 97

8

xx

x

x

xx

+=− = + ⇔ − + = ⇔ −=

��បGប�ធ@បនDង5កqខ:r (*) ន�ង (**) , ម� �).5 !&ន�ន��UគM�

11 73 15 97;

8 8S

− − =

�. �យ/ង��យបI% ក"J ម� �).5 !ម�ន&នU'ច�ន�នគ�"�ផpងព� 0; 0(0; ; 0; 0) �ទ�

zប&ផ7�យមក# �ញ, &នU'ច�ន�នគ�" ( (0;0;0;0;0); ; ; ; )x y z t u ≠ �

=ង 0 0 0 0 0; ; ;( ; )z tx y u 'Uគ�"ច��Z� 0 0 0 0 0x y z t u+ + + + 'ច�ន�ន# �ជ%&ន�*ចប�ផ���

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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=ង 0 15x x= �

ជ�ន�ច*5ម� � �យ/ង;ន 5 5 5 5 51 0 0 0 05 2562 125 05 ux y z t+ + + + = �

ព��ន� �យ/ង)ប�'Sញ;ន 0y )ចក�ច"នDង 5� =ង 0 15y y= , ជ�ន�ច*5ម� �,

�យ/ង;ន� 5 5 5 5 51 1 0 0 01 625 5 25 52 0y z t ux + + + + = �

ព��ន� Sញ;ន 0z )ចក�ច"នDង 5 � ��យបI% ក".*ចA� , �យ/ង;ន 0t ន�ង 0u �ទQ)�

)ចក�ច"នDង 5�

=ង 0 1 0 1 0 15 , 5 , 5z t t u uz = = = , �O/យច�ង�� យ�យ/ង;ន�

5 5 5 5 51 1 1 1 15 25 125 625 0y z tx u+ + + =+ �

.*ច�ន� �យ/ង;នU'ច�ន�នគ�" 1 1 1 1 1; ; ;( ; )z tx y u �ប"ម� �ច��Z�

1 1 1 1 1 0 0 0 0 00 x y z t u x y z t u< + + + + < + + + +

ផ7�យព� ���ជ/យក 0 0 0 0 0; ; ;( ; )z tx y u � .*ច�ន� �zប& គMខ� �O/យ�យ/ង;នបIg

��e#;ន��យបI% ក"�

��បមក, ម� �).5 ! &នU'ច�ន�នគ�")�ម�យគ�"គM 0; 0(0; ; 0; 0)�

1. ABCD 'ច��� :Z� យ

||AE DE

AD BCAB DC

=⇔ ⇔

. . 1AE DC FB

AB DE FC=⇔

, ,BA D FC E⇔ �បពBA� ��ង"ច�ន�ចម�យ (=ម�ទD-�បទ Ceva )

, ,E T F⇔ ��"��ង"ជ���

E. Fន�#�-នj# �មiព Cauchy Schwarz− ច��Z� 6ច�ន�ន�

; ; ; ; ;pm n

m n pn p p m m n+ + +

�យ/ង;ន ( )2 2 2( ) ((

()

)1

)

m n p m n pm n p

n p p m m n n p p m m n

+ + + + + + + + + + + +

=ម# �មiព Nesbitt

�យ/ង;ន 2 2

3(2)

2

m n p

n p p m m n

+ + + + + ≥

B

A

C

D

E

F

T

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 76

ព� (1) ន�ង (2) �យ/ង;ន 2 2 2

9

( ) ( ) ( ) 4( )

m n p

n p p m m n m n p+ +

+ +≥

+ + +

មiព�ក/�&ន�ព5 m n p= = �

K. ក�:� n 'ច�ន�នគ�"# �ជ%&នគ*� ប,- �:5"ក��ង�ប&:#�ធ�)ចកនDង n ;ន)ចក�ចញ'

12

n + �កmម.*ច�ង�� ម� { } { } 2 20 , 1; , ..., ;

2 21 ,

2

n nn

n− +−

ច��Z� n 'ច�ន�នគ�"# �ជ%&នគ*, �យ/ង;ន 2 22 2

n n + = + � =ម�ទD-�បទ Dirichlet , &ន

89ង��ចព��ច�ន�ន ក��ង 22

n + ច�ន�នគ�",កL��យ ).5&ន�:5"�ព5)ចកនDង n ;ន

4���2ក��ង�កmមម�យ ក��ងច��,ម�កmម�ង�5/, គMJ ព�ក?&នផ5ប*ក � Mផ5.ក)ចក�ច"

នDង n � .*ច�ន� �យ/ង;ន បIg ��e#��យបI% ក"�

ក�:� n 'ច�ន�នគ�"# �ជ%&ន�� ប,- �:5"ក��ង�ប&:#�ធ�)ចកនDង n ��e#;ន)ចក

�ចញ' 1

2

n + �កmម.*ច�ង�� ម� { } { } ...1 1

0 , 1; 1 ; ;2 2

,n n

n− + −

ច��Z� n 'ច�ន�នគ�"# �ជ%&ន�, �យ/ង&ន� 1 12 2 1

2 2 2

n n n− + + = + = +

=ម�ទD-�បទ Dirichlet , &ន89ង��ចព��ច�ន�នក��ង 11

2

n + + ច�ន�នគ�",កL��យ ).5&ន

�:5"�ព5)ចកនDង n ;ន4���2ក��ង�កmមម�យ ក��ងច��,ម�កmម�ង�5/, គMJ ព�ក?

&នផ5ប*ក � Mផ5.ក )ចក�ច"នDង n �

.*ច�ន� �យ/ង;នបIg ��e#��យបI% ក"�

N. ��ជ/�� /����យក*F����ន Oxy ច��Z� (0;0)O 'ច�ន�ចក,- 5�ប" BC , កន3�ប67 �" Ox

��|�នDងកន3�ប67 �" OC , កន3�ប67 �" Oy ��|�នDងកន3�ប67 �" OA�

=ង 2 , hBC a OA= = � �យ/ង;ន ( ;0(0 ),; ), ( ; , ;2 2 6 2

;0),a

B a C a M Gh a h

A h − −

=ង 0(0; )I y

�យ/ង;ន 0;2 2

a hyIM = − −

�, ( ; )AB a h= − −�

=មប�^ប", �យ/ង&ន�

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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2 2

0. 02

aIM AB IM AB

hy

h

−+ ⇔ = ⇔ =� � � �

�យ/ង;ន� 2 2 2

0; , ; ,2 6 2

aIG

h a aI

h h

− =

2 2

.3

; ; 02 2 4 4

CM IG Ca h a a

M− − + ==

=

� � �

.*ច�ន� IG CM⊥� �

,គMJ IG CM⊥ �

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១៥

�. �����យម� � ( )( )2 22 2 2 1 24 6 6 7x x x xx x+ + − − +=+ + +

�. �គ !���� : ABC �ផ7GងH7 �"ទ�6ក"ទ�នង�ង�� ម�

( )( ) ( ) ( ) ( ) ( )2 2cos 2 2cos 2 2cos

361 cos 1 cos 1 cos 1 cos 1 cos 1 cos

A B C

B C C A A B

+ + ++ + =− − − − − −

ក�:�"^ង�ប"���� : ABC �

1. ក�:�"�គប"��C5�ប" a �./ម0� !ម� � 2 2

1 11

a

xyx y+ + = &នU'ច�ន�នគ�"# �ជ%&ន.

E. �ក�គប"ប,- ពO�o ( )f x �ផ7GងH7 �"� 22 ( ) (1 ) ( )mf x f x mx− = ∈+ ℝ ន�ង (1) 2f = �

K. �គ ! , ,a b c 'ប,- ច�ន�នគ�"ធមn'��ខ�ព�*ន! ន�ង�ផ7GងH7 �"5កq:r 100a b c+ + =

�ក��C5ធ�ប�ផ�� ន�ង��C5�*ចប�ផ���ប"ក�នyម P abc= �

N. �គ !�ងBង"ផa�� O \�Dក��]���� : ABC , ប,- ក�ព" 0 0 0, ,BBA CCA �"A� ��ង"

H � =ង 1 1 1, ,A B C ��@ងA� 'ច�ន�ចឆ3��A� នDង 0 0 0, ,A B C �ធ@បនDងច�ន�ចក,- 5�ប"

ប,- �ជmងន�ម�យ�� ��យបI% ក"J �គប"ច�ន�ចក,- 5�ប"ប,- �ជmង���� :

1 1 1A B C 4���2�5/ប67 �" , ,OA OB OC � '()&'()&'()&'()&

B

A

C x

y

M G

O

I

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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ចំេល�យ

�. 5កqខ:r � 2 4 6 0

2 1 0

2 1

2

xx

x

x

+ + ≥⇔ ≤ −

− − ≥

ម� �).5 ! 3 យ�P'�

( )2 4( 2) 2 2 16xx x x+ + −+ −+

( ) ( )2 22 2 1 . 2 16 24 4 6x xx x xx+ + + += + − − − − −

( ) ( )2 22 2 1 . 2 2 1 24 6 4 6 0x x x x xx x⇔ + + ++ − − + − − − − − =

2

2

4 6 (1)

2 4 6 2 1 2 0 (2)

2 2 1 0x

x

xx

x x x

+ +⇔

+

+ − − =

+ − − − − − =

2 4 6 02

(1)2 1 0

x

x

x+

+ + =⇔

− − =

(�បព<នQAn នច��5/យ)

22( 2)(2) 2( 2 1) 2 2 1x xx x⇔ + − = + + −−+ + −

=ង� 1 0

2

2

v

u x

x

≥= +

= − −

�ព5�6� (2) 3 យ�P'� 2 222 vu u v+ = +

2 2 2 2

0 0

2 2 ( ) ( ) 0u v u

u v u vu v

v u v

+ + =

≥ ≥⇔ ⇔ ⇔

+ = + − =

.*ច�6� 2

22 1 2

20

12 1 ( 2)

5

1xx

xx

x x xx

x

+ − − = + = −

≥ −≥

⇔ ⇔ ⇔= −− − = +

= −

.*ច�ន� �ន��U�ប"ម� �� { }1T = − �

�. + �យ/ង&ន� ( )22 2 2 ;3

, ,1

a a bb c ac b c+ + ≥ + ∀+ ∈ℝ

.*ច�6�

2 2cos 2 2cos 2 2cos

(1 cos )(1 cos ) (1 cos )(1 cos ) (1 cos )(1 cos )

A B CM

B C C A A B

+ + += + +− − − − − −

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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2 2

2 2 2 2 2 2

cos cos cos2 2 2

sin .sin sin .sin sin .sin2 2 2 2 2 2

A B C

B C C A A B= + +

2

cos cos cos1 2 2 23 sin .sin sin .sin sin .sin

2 2 2 2

(

2

1)

2

A B C

B C C A A B

+ +

)���យ cos cos

2 2cot cot , cot cot ,2 2 2 2sin .sin sin .sin

2 2 2 2

C AA B B C

A B B C+ = + =

cos

2cot cot2 2 sin .sin

2 2

BC A

C A+ = �

Sញ;ន cos cos cos

2 2 2 2 cot cot cot2 2 2sin .sin sin .sin s

(2in .sin

2 2 2 2 2 2

)

A B CA B C

B C C A A B + + = + +

+ �យ/ង��យបI% ក"J cot cot cot 32 2 2

3 (3)A B C+ + ≥

ព��'.*ច�ន�, �យ/ង&ន�

2

cot cot cot2 2 2

A B C p a p b p c p p

r r r r S

− − −+ + = + + = =( )( )( )

p p

p a p b p c=

− − −

)�=ម# �មiព 3

( )( )( )3 3

:3

p ppCauchy p a p b p c ≤ − − − =

�6� .3 3cot cot cot 3 3

2 2 2

p pA B C

p p+ + =≥ (បIg ��e#��យបI% ក")

+ ព� (1), (2) ន�ង (3)Sញ;ន ( )21. 2.3 3 3 (4)

36M ≥ =

.*ច�ន� =មប�^ប"�បoនគM 36M = �6� (4) �ក/�&នIw " "= �ព5 A B C= =

.*ច�6� ���� : ABC '���� :ម<ងp�

1. zប&J ( );x y 'Uគ�"# �ជ%&ន�ប"ម� � 2 2

1 11 (1)

a

x xy y+ + =

=ង 1 1( , ) ,d GCD x y x yx dyd=⇒ == ច��Z� 1 1, ) 1( yx =

Page 81: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 80

�យ/ង;ន 2 2 2 2(1) axy x yy x⇔ + + =

2 2 2 21 1 1 1 1( 1) ( ) 1)( ) (y y a y y axx y x dyx⇔ + ⇔ == − + − .

��យ 1 1, ) 1( yx = �6� 21 1 11)( ydy y− ⇒⋮ '��)ចកCន 1 ��Z� y 'ច�ន�នគ�"# �ជ%&ន�

�6� 1 11 1y x= ⇒ = � .*ច�6� x y=

ជ�ន�ច*5 (1) �យ/ង;ន� 22a x+ =

.*ច�ន� �./ម0� ! (1)&នU'ច�ន�នគ�"# �ជ%&ន គM 2a + ��e#'ច�ន�ន ���;ក.�

E. ��យប,- ក�នyម�� មIw f '.M��កទ�ម�យ� 1;x x− , FងR�ង�- �'ក�នyម&ន

.M��កម�ន�5/ព� 2 �6� ( )f x &ន.M��កម�ន�5/ព� 2

.*ច�ន� ( )f x &ន^ង� 2ax bx c+ +

�ព5�6� 2 2 22 ( ) (1 3 ) 3) ( 2b a x a b c mf x f x mx a xx⇔ + − + + + =+ − =

��យផ7DមFងRង�ង �យ/ង;ន�បព<នQ� 332

2 03

3 0

3

ma

a mm

b a b

a b cm

c

==

− = = + + = = −

.*ច�ន� ( )2 21

( )3

f x mx mx m+= −

��យ 1(1) 2 .2 2

33mf m= =⇔ ⇔ =

.*ច�6� 2( 1) 2f x xx += −

�កជ�ន�ច*5 �យ/ង�ឃ/ញJ 2( 1) 2f x xx += − �ផ7GងH7 �"5កqខ:r �បoន�

ផ7�យមក# �ញ� zប&J &នពO�o ( )g x ម�យ ម�ន.*ចA� នDង ( )f x �O/យ�ផ7GងH7 �"��:/ �

�បoន �ព5�6� 0x∃ 89 ង, ! 0 0(( ) )f g xx ≠

�យ/ង;ន� 2

0 0 0 20 0 0 02

0 0 0

) (1 )) 2 1 ( )

) ( ) (

2 ((

2 (1 1 )

g xg

g x xxx

gx f

xxx

g x

+ − =⇒ = + − =

− − + = (ផ7�យព� �zប&)

.*ច�ន� ពO�o 2( 1) 2f x xx += − 'ពO�o)�ម�យគ�").5��e#�ក�

K. ��យម�ន�ធB/ !;�"5កq:�ទ*�P, �យ/ងzប&J a b c≥ ≥ , Sញ;ន 34a ≥ �

� �ក��C5ធ�ប�ផ��

Page 82: Vnmo 30 4-2011-grade 10

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Page 81

�ព5�6� �យ/ង&ន�

3 33 34 34 34( )

333 .34 .34 33.3

34a b

a b c a b c ac

+ + + + −=≤ ≤

Sញ;ន 333 .34abc ≤

.*ច�ន� ��C5ធ�ប�ផ���ប" 233 .34abc = ទទ�5;ន�ព5 3334,ba c= == �O/យន�ង�គប"

ច��"�ប"?�

� �ក��C5�*ចប�ផ��

+ �ប/ 1c > Sញ;ន 2c ≥ ន�ង 2b ≥ , �ព5�6� �យ/ង;ន

34.2.2 136 98P abc ≥ = >=

+ �ប/ 1c = , �6� 99a b+ = Sញ;ន 50a ≥

i �ប/ 1b > , Sញ;ន 2b ≥ , �ព5�6��យ/ង;ន 50.2.1 100 98P abc ≥ = >=

i �ប/ 1b = �6� 98a = , �ព5�6� 98.1.1 98P abc= = = �

.*ច�ន� ��C5�*ចប�ផ���ប" 98abc = �ព5 , 198 ba c= == ន�ង�គប"ច��"�ប"?�

N. .�ប*ង, ��យ# �ធ�ប�)បក#� �ចទ<� OC�

=ម OA�

ន�ង OB�

, �យ/ង��យបI% ក";នមiព�

sin 2 . sin 2 .sin 2 . 0 (*)OA B OB C CA O+ + =� � � �

ព63 � 1OB�

ន�ង 1OC�

=មប,- គ*#� �ចទ<� ,OC OA� �

ន�ង ,OA OB� �

, �យ/ង;ន�

1

cos . cos .a C OC c A OAO

bB

+=� �

1

cos . cos .b AOA a B OBOC

c

+=� �

ប*ក��មនDង (*) Sញ;ន#� �ចទ<� 1 1u OB OC= +� �� &នទ�.*ចA� នDង OA

� � M ច�ន�ចក,- 5�ប"

1 1B C 4���2�5/ប67 �" OA� ព��6� �យ/ង;ន បIg ��e#��យបI% ក"�

បក��យ.*ចA� ).� ច��Z�ប,- ក�:� ).5�25"�

'()&

Page 83: Vnmo 30 4-2011-grade 10

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Page 82

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១៦

�. �����យ�បព<នQម� �� 2 2 2

2 2

2 3(1)

2 1

y z xy xz zy

x y yz xz xy

x + + + −

− =

+ + − − = −

�. �គ ! 1 2 17, , ...,aa a ' 17ច�ន�នព��ខ�A� ព�ម�យ�Pម�យ� ��យបI% ក"J �គ)�ង

��ជ/យក;នព��ច�ន�ន ,j ia a �ចញព� 17ច�ន�ន�6��./ម0� !�

0 4 2 2 11

j i

i j

aa

a a< < −

−−

+

1. =ង ,, , ,,a b chR hr h p ��@ងA� ' ��ងBង"\� Dក��], ��ងBង"\� Dកក��ង, ប,- ក�ព"គ*

�ចញព� , ,A B C ន�ងកន3�ប� �&���ប"���� : ABC �

បfg ញJ 2 2 2 ( 2 )a b cr R h h h rp + = + + − �

E. �គ !�បព<នQ�

2 2

2 2

4

9

6

y

z v

xv yz

x + =

+ =+ ≥

, �កច��5/យ�ប"�បព<នQ�./ម0� !ក�នyម P xz= &ន��C5

ធ�ប�ផ���

K. �គ&ន 2012ក*ន&ន").5�កmងក��ង 1006�ទmង, �ទmងន�ម�យ�&នក*ន&ន"ព��កt5�

ប67 ប"ព�ម�យCថ�� �គH3 "ប-*�ទ�=�ង�ប"ក*ន&ន" 89ង, ! ម�ន&នក*ន&ន"ព��កt5

,).5o3 ប"�2ក��ង�ទmង'ម�យA� ព��ព5ម�ន 4���2ក��ង�ទmង'ម�យA� ម-ង�ទ@��ទ�

��/&ន�យ��ព5��ច/នប�ផ��ប9�6n នCថ� ក��ង ��ធB/.*ច�ន�;ន?

N. �ក k �./ម0� !�បព<នQ�ង�� ម&នច��5/យ��ច/នប�ផ��� 2 2

1 (1)| 1|

(

1|

2

|

)

x y

x y k

=

+

− + +

=

'()&'()&'()&'()&

Page 84: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 83

ចំេល�យ

�. �យ/ង&ន 2 2

2

( ) 3 0(2)

(

( )(1)

) ( ) 1 0

z x y z

x

x y

y z x y

− + + − =⇔

− −

+ − + =

=ង 2

2

u vxu x y

v x y u vv

+ == + = −

⇔− =

�បព<នQ (2) 3 យ�P' 2 2

2

3 0(3)

1 0

zu z

v

u

zv

− + − =−

+ =

�បព<នQ (3) &នច��5/យ 2

2

0 42

0 4

u

v

zz

z

∆ ≥ ≤

⇔ ⇔ ⇔ = ±

∆ ≥ ≥

ច��Z� 2z = ;ន 1(3) 1

0

xu v

y⇔ ⇒

== = =

.*ច�ន� �បព<នQ).5 !&នច��5/យ (1;0; 2)

ច��Z� 2z = − ;ន 1(3) 1

0

xu v

y⇔ ⇒

= −= = − =

.*ច�ន� �បព<នQ).5 !&នច��5/យ 0 2( 1; ; )− −

��បមក �បព<នQ).5 !&នច��5/យព�� (1; ( 1;0; 2); 0; 2)− −

�. ��យម�ន�ធB/ !;�"5កq:�ទ*�P �យ/ងbចzប&J 1 2 17...aa a< < <

=ង ; 1, 2, ...,17tan ;2 2i i ia iv vπ π= − < =<

=ម5កq:��ក/ន�ប"Fន�គមនj tany x= ក��ងច�63 � ,2 2

π π −

�6�ព� 1 2 17...aa a< < < Sញ;ន 1 2 17 1...2 2

v vv vπ π π< < < << < +−

ប,- ច�ន�ច 2 3 17, , ...,vv v )ចកFង̀�" [ ]1 1;v v π+ �P' 17Fង̀�" ក��ង�6�&ន89 ង��ចFង̀�"

ម�យ).5&ន�ប)#ងម�ន�5/ព� 17

π �

)a �ប/&ន i ម�យ ).5 1 16i≤ ≤ 89ង, ! 1017i iv vπ

+ − ≤<

�6� 10 tan( tan17 16

) tani ivvπ π

+ − ≤< < �

Page 85: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 84

��យ 2

2 tan8tan

4 1 tan 18

ππ

π=− =

Sញ;ន 2

2 tan16tan 2 1, tan 2 1 tan 4 2 2 1

8 8 161 tan16

ππ π π

π= − = = − = − −−

�ព5�6� 1 11

1 1

tan0 tan( 4 2 2 1

1 tan t

tan)

an 1i i i i

i ii i i i

v av

v a

v

v

av

a+ +

++ +

< = < − −+

−−+

−=

យក 1j ia a += �យ/ង;ន បIg ��e#��យបI% ក"�

)b �ប/ 1 17017 16

v vπ ππ+ − ≤< <

�6� [ ]1 17 1 170 tan ( tan )1

) n(6

tav vv vππ+ − << =−

�ព5�ន� �យ/ងយក 1 17;j ia aa a= = �យ/ង;ន បIg ��e#��យបI% ក"�

1. ��យ 2 2 22 ) 2( a b c

S S SR h

a bh

ch R+ + = + +

4ab bc ca

RS ab bc caabc

+ + = = + +

��យបIg ��e#��យបI% ក" មម*5នDង 2 2 4r Rr a b cp b c a+ + = + +

មu9ង�ទ@� 2

2 tan2sin

1 tan2

A

AA

=+

, ជ�ន� tan ; sin2 2

A r aA

p a R= =

�យ/ង;ន ( )3 2 2 22

2

2

4 02

14 (

(

2 2)

)

p

ra p a

a p a pRrr

a r RR

p a

r⇔ − + + +−= − =+

.*ចA� ).� ( )3 2 2 22 4 0 )4 (3b p b pRrpb r Rr+ =− + −+

( )3 2 2 22 4 0 )4 (4c p c pRrpc r Rr+ =− + −+

ប,- ទ�6ក"ទ�នង (2), (3), (4) បfg ញJ ,,a b c �ទQ)�'ប,- U�ប"ម� �

( )3 2 2 22 4 4 0pX r Rp pRrrX X −− + + =+

Page 86: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 85

�6� ( )3 2 2 2 4 ( )(4 )(2 )X p X ppX r R Rr X a X b X cr− + + − −≡ −+ − ≡

3 2( ) ( )a b c X ab bc cX a X abc− + + + + + − ��យផ7Dម�មគ�:ប,- ��).5��e#A� �2FងRង�ង�យ/ង;ន (1)�

E. =ង 2sin ,2cos , 3cos , 3sinx a z by a v b= == = ច��Z� [0 2 ], ;a b π∈

�ព5�6� 6 6(cos sin sin cos ) 6 i 66s n( )xv yz a b a ab b+ +≥ ⇔ + ≥ ⇔ ≥

# �មiពច�ង�� យ ព��ក��ង�ព5).5 ( )2

1a bπ+ = )�ប9��,c �

ក��ងក�:� �ន� គM [ ]6cos cos 3 cos( ) cos( ) 3cos( )P xz a b a b a b a b= = = + + − = −

&ន��C5ធ�ប�ផ�� �ព5 cos( ) 1 0a b a b− = − =⇔

ប*ក��មនDង (1) Sញ;ន 4

a bπ= = ��e#A� នDងU 3 2

2;2

zx y v= == = �

K.

ង"ពO�� :ន�យ<� 2011�ជmង \� Dកក��ង�ងBង"� =ង ផa����យ 2012 �O/យប,- ក�ព*5

��@ងA� ��យ 2,3, 4, ..., 2009, 2011, 0, 2011�

ព�ន��! � 1 2012− , =ម5កq:�ពO�� :ន�យ<� �6�&ន 1005)ខpធ�*ច�ង�� យ )កងនDង

��6� : 3 2010; 4 2009; ...;12 2011; 006 1007− − − − �

.*ចA� ).� ព�ន��! � 2 2012− , =ម5កq:�ពO�� :ន�យ<� �6�&ន 1005)ខpធ�*ច�ង

�� យ)កងនDង ��6� : 4 2011; 5 2010; ...;10063 1; 1008− − − − �

�ច�)�បន-�ធB/.*ច�ន��O*�.5" � 2011 2012− , =ម5កq:�ពO�� :ន�យ<� �6�&ន

1005)ខpធ�*ច�ង�� យ)កងនDង ��6� 2 2009; 3 2008; ...;: 1 2010; 1005 1006− −− − �

��យ)ផ�ក�5/5ទQផ5�ង�5/, �យ/ងbចបfg ញព���ប@ប��@ប&ន"=ម��:/ ��បoន

.*ច�ង�� ម�

2010

2009

2011 1 2 3

4

2012 2012

2011

2010

1 2 3

4

5

2012

2011 1 2 3

4

5

6

Page 87: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 86

បង"�5ខ5��ប"ព� 1 .5" 2012�P !&ន"S�ង 2012កt5�

Cថ�ទ��� ��@បច*5ប,- �ទmងន*#ប,- គ*&ន"�ង�� ម� 2 2011;1 2012; 3 2010;−− −

...;1004 2009; 6 1007− − ��e#A� នDងក�:� � 1 2012− �

Cថ�ទ��� ��@បច*5ប,- �ទmង ន*#ប,- គ*&ន"�ង�� ម� 3 1;2 2 4012 11;; 20− −−

...;1005 2010; 7 1008− − ��e#A� នDងក�:� � 2 2012− �

�ធB/.*ចA� �O*�.5"Cថ�ទ� 2011� ��@បច*5ប,- �ទmង ន*#ប,- គ*&ន"�ង�� ម�

1 2010; 2 2009; 3 2008; .2011 20 ..12; ;1005 1006− − −− − ��e#A� នDងក�:� � 2011 2012− �

�;ក.' ម�ន&ន&ន"ព��កt5,�ក"'ន"A� �ទ ��Z�ប,- )ខpធ�* ន�ង ��ង�5/�ទQ)�

�ផpA� �O/យម�នbច��ច/ន'ង 2011Cថ���@បប�a* 5�ទmង�ទ ��Z�ក*ន&ន"ន�ម�យ� bច��@ប

ប�a* 5�ទmង'ម�យនDងក*ន&ន" 2011កt5�25"ប9��,c ��

.*ច�ន� &ន89ង��ច/ន 2011Cថ�).5bច��@ប&ន"=ម��:/ ��បoន�

N. �).ង�2�5/ប3ង" ( )Oxy , ប,- ច�ន�ច ( , )M x y �ផ7GងH7 �" (1)' �� ABCD ).5&ន

ក*F����ន ប,- ក�ព*5គM (1;0); (2; 1); (1; 2) )(0; 1 ;B C DA − −− �

�ឃ/ញចt"J �យ/ង�Aន")���e# �ព�ន��!ច��Z� 0k > � ប,- ច�ន�ច�ផ7GងH7 �" (2)

'�ងBង"&នផa��'គ5"����យ, � k �

ង" OH DC⊥ , �ឃ/ញJ 3 2

2OH =

�ងBង"ផa�� O � OH ប9� CD �O/យ �"

�ជmងS�ងព�� ,AD BC (�ជmងន�ម�យ���ង"

ច�ន�ចម�យ)� �យ/ង;ន 5OD = , ��Z�

�ងBង"ផa�� O , � OD �"=ម)�ព��ក�ព*5 ,C D �

ព��6� Sញ;ន ច�ន�នច�ន�ច�បពB�ប"�ងBង" ន�ងប,- �ជmង ����ច/នប�ផ��គM 4 �ព5�

3 2 95 5

2 2k k⇔< < < < �

'()&

O

A

B

C

D H

y

x

Page 88: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១៧

�. �����យ�បព<នQម� �� 2 2 3 5 7

3 5 2 3 1

x y x y

x y x y

+ + − − =

− − − + − =

�. ��យបI% ក"J� �ប/ ,m n 'ប,- ច�ន�នគ�"ធមn'���ផ7GងH7 �"ទ�6ក"ទ�នង� 2 23 4m nm n+ = + �6� m n− ន�ង 4 4 1m n+ + �ទQ)�'ច�ន�ន ���;ក.�

1. �គ ! ABC∆ &ន , ,Ca A B cB bC A= == � ��យបI% ក"J� ច��Z��គប"ច�ន�ច M

4���2ក��ង���� : ABC �យ/ង;ន�

( ) ( ) ( )2 2 2 6a bc MA b ca MB c ab MC abc− + − + ≤−

E. �គ ! , ,a b c 'ប�ច�ន�ន# �ជ%&ន &នផ5គ�:�n/ 1� ��យបI% ក"J�

4( 1( )( ) )( )a b b c c a a b c≥ + ++ −+ + .

K. �គ ! n ច�ន�ច�2ក��ងប3ង"� ��យបI% ក"J ច�ន�នគ*ច�ន�ច).5&ន�ប)#ង�n/ 1គMម�ន

�5/ព� 322

4 2

nn+ �

N. �គ !ប�ច�ន�ច , ,A B C 4���2�5/ប67 �"ម�យ ( B 4���2ច�63 � Aន�ងC )� �ងBង" ( )I ម�យ

�"=ម Aន�ង C ( I ម�ន4���2�5/ប67 �" AC )� ប67 �"ប9��ប" ( )I ��ង" Aន�ងC

�"A� ��ង" P � =ង D 'ច�ន�ច�បពB�ប" PB នDង ( )I � ��យបI% ក"J ច�ន�ច

�បពB�ប"ប67 �"ព��ម�� ADC នDង AC ម�នb�<យនDង�ងBង" ( )I � '()&'()&'()&'()&

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ចំេល�យ

�. =ង 2 3 , 5 , 2 3u x y v x xwy y= + − == − + −

�បព<នQម� �).5 ! 3 យ�P' 2 2 2

1

1

2 7

3

4 7v w

u v

v w

u +

+ = − =

+

=

� M 2 2 24 17

732

3 1 1

2v

vu

u

w v v

wu w

− = = = − = =

⇔=

+ +

ព��6�, 2 3 9 3

5 1 1

x y x

x y y

+ = = − − = =

�. �យ/ង&ន� 2 2 2 2 24 ( ) ( )3 4m mm n n n m n m+ = + ⇔ − + − =

2( )(4 4 1) (*)m n m n m− + =⇔ +

=ង d '��)ចក��មធ�ប�ផ���ប" m n− ន�ង 4 4 1m n+ +

�6� (4 4 1) 4( )m n m n+ + + − )ចក�ច"នDង 8 1d m⇒ + )ចក�ច"នDង d �

មu9ង�ទ@�, ព� (*) �យ/ង&ន� 2m )ចក�ច"នDង 2d m⇒ )ចក�ច"នDង d �

ព� 8 1m + )ចក�ច"នDង d �6� m )ចក�ច"នDង d �យ/ង;ន 1)ចក�ច"នDង 1d d⇒ = �

.*ច�ន� m n− ន�ង 4 4 1m n+ + 'ប,- ច�ន�នគ�"ធមn'��ប[ម�?ងA� , �ផ7GងH7 �" (*)

�6�ព�ក? �ទQ)�'ប,- ច�ន�ន ���;ក.�

1. =ង I 'ផa���ងBង"\� Dកក��ង���� : ABC , �យ/ង;ន

2(0 ) 0a aIA bIB cIC IA bIB cIC+ + = ⇒ + + =� � � � � ��

2 2 2 2 2 2 2 2 2 0( )IA b IB c IC abIAIB bcIBIC Ca caIAI⇔ + + + + + =�� �� ��

2 2 2 2 2 2 2 2 22 )(a ab IAIA b IB c IC IB c⇔ + + + ++ +

2 2 2 2 2 2) 2( (2 ) 0IC a ac IA ICbc IB b+ − + + − =+

2 2 2) ( ) 0( )( bIB cICa b abc a b cc aIA⇔ + + − + ++ =+

2 2 2bIBaI cIC abcA⇔ + + =

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ច��Z��គប"ច�ន�ច M 4���2ក��ង���� : ABC �យ/ង&ន�

2 2 2 2 2 2 2( )bMB cMC aIA bIB cIC aaMA a c bcb MI+ + ++ + = + + ≥

=ម# �មiព Bunyakovski �យ/ង&ន�

2 2 2( ) ( () )a bc MA b ca MB c ab MC− + − + −

2 2 2(1 1 1) ( ) ( ) ( )a bc MA b ac MB c ab MC ≤ + + − + − + −

2 2 2)3 3 ( 3(3 ) 6bMB cMCabc aMA abc abc abc = − − = ≤+ +

E. zប&J a b c≥ ≥ � .*ច�ន� 1a ≥

ព�ន��! , ) ( )( )( ) 4(( , 1)b c a b b c c a a bf ca = + + + − + + −

�យ/ង&ន 2, ) ( , , ) ( ) ( )( ) 4 0( , b c f a bc bc b c a b a c aa − = − + + + − ≥

��យ ( )( ) .4 4.4 4a b a c ab ac a abc a+ + = =≥ ≥

=ង t bc= , �យ/ង;ន�

2 4 3 22

1( , , ) , , ( 11) 0( 2 )f a bc bc f t t t t t

tt = = −

+ − + ≥

# �មiពច�ង�� យគMព��, ��Z�=មប�^ប" t bc= �6� (0;1]t ∈

.*ច�ន� , ) ( )( )( ) 4(, 1) 0(f a b c a b b c c a a b c= + + + − + + − ≥

.*ច�6� �យ/ង;ន� 4( 1( )( ) )( )a b b c c a a b c≥ + ++ −+ + �

K. ក�:�" n ច�ន�ច ' n ក�ព*5�ប"� ប G ម�យ� =ង { }1 2; ; ...; nV vv v= '�ន��ប,- ក�ព*5

�ប"� ប G � ព��ក�ព*5 ��e#;ន��J�2ជ��A� �ប/�ប)#ង�?ងព�ក?�n/ 1� �យ/ង&ន

��ង̀�� ផ5ប*កប,- 5��ប"�ប"�គប"ក�ព*5�n/ 2.ងច�ន�ន�ជmង�ប"� ប គMJ

1 2) ( ) ... ( )2 ( nde d v v d v+ + += ច��Z� e 'ច�ន�ន�ជmង�ប"� ប, )( id v '5��ប"�ប"ក�ព*5

( 1; 2; ...; )iv i n= �

=ង iC '�ងBង"&នផa��គM iv ��n/ 1� ផ5ប*កប,- ច�ន�ច�បពB�ប"�ងBង"ព��,កL

��យក��ងច��,ម n �ងBង"�6� ម�ន�5/ព� 2 (. )2 1n n nC = −

មu9ង�ទ@�, �ប/ iv �2ជ��A� នDងព��ក�ព*5 ;j kv v �6�គM i j kv CC∈ ∩ � .*�ច��, iv ��e#;ន\�"

ទ�ក.*ច' ច�ន�ច�បពB�ប"�គប"ប,- �ងBង" �O/យ��e#;ន^ប" 2( )

)( )

2

) 1( (i

i id v

d vvC

d −=

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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.*ច�6�, �យ/ង;ន� 1 2

2 2 2( ) ( ) ( )... ( 1)

nd v d v d vC C C n n+ + + ≤ − (=ម 1)

1 1 1 2 (( (

2 2

)( ( ) 1))( ( ) 1) )( ( ) 1)...

2( 1) (2)n nd vd v d vd v d v

nd v

n−− + ≤ −+ − +⇔

[ ]2 2 21 2 1 2( ) ( ) ... ( ) ) ( ) ... ( ) 2 ( 1)(n nv d v d v d vd d v d v n n − ⇔ + + + + + + ≤ −

=ម# �មiព�យ/ង&ន�

[ ]2

22 2 21 2 1 2

1 4( ) ( ) ... ( ) ( ) ( ) ... ( )n n

ev d v d v dd v d v d v

n n+ + + ≥ + + + =

�6�ព� (1) ន�ង (2) Sញ;ន

2

2 242 22 ( 1) ( 1) 0n n ne

ee

nne n≤ − ⇔ − − − ≤−

33 2 32

8 8 2

4

7

4 4 2

n n nn n ne n

+ + = +−⇒ ≤ ≤

បIg ��e#��យបI% ក"�

N. # �ធ�ទ���

��ជ/យក����យក*F����ន Oxy .*ច�*ប�

zប&J ក*F����នច�ន�ច ( ;0), (1( 1; 00 , )) ;B b CA −

�O/យក*F����នផa�� (0; )I p− �

ម� ��ងBង" 2 2 2: () ) 1( y px pI + + = +

ក*F����នច�ន�ច 1(0; )P

p , ក*F����នច�ន�ច 2(0; 1 )G p p− − +

ម� �ប67 �" 10:

1PB x y

pb p+ − =

�ព5�6� ក*F����ន 2 2 2 2 2 2

2 2 2 2

) )(1(1 (1 (1 (1;

1 1

) ) )(1 )p p b pD

p

b pb b

p

b p

b b

+ + − + + +

− −

+ −

.*ច�ន� �យ/ង;ន� 2

2

1

1

pDP

DB p b

+=

)���យ 2

22

11 11

p pGO p

GP pp pp

+ += =

++ + +

( 1;0)A − E

B

D

(1;0)C

x

y

(0;1/ )P p

(0; )I p−

O

G

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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�ព5�6� �យ/ង;ន� 2

2 2 2

1 1. .

1 1 1

pOE GO PD p

EB PG BD p p b b

+= = =

+ − −

.*ច�ន� ច�ន�ច E ម�នb�<យនDង p � M ( )I �ទ�

# �ធ�ទ���

=ង ,E G ��@ងA� 'ច�ន�ច�បពB�ប"ប67 �"ព��ម�� ADC នDង AC ន�ង ( )I �

��យ���� : PAC ម;���ង" C �6��យ/ង;ន� sin

sin

AB APB

BC CPB=

���� : GAC ម;���ង" G �6��យ/ង;ន� sin

sin

AE AGD

EC CGD=

=ម�ទD-�បទ 'Ceva s ច��Z����� : PAC ន�ង D �យ/ង;ន�

sin sin .sin

sin sin .sin

APB PAD DCA

CPB PCD DAC=

)���យ PAD AGD ACD∠ = ∠ = ∠ ន�ង (2)PCD CGD CDB∠ = ∠ = ∠

ព� (1) ន�ង (2)�យ/ង;ន 2

2

AB AE

BC EC= �

.*ច�ន� E ម�នb�<យនDង ( )I �ទ�

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១៨

�. �����យម� � ន�ង�បព<នQម� ��

2 2

3 2 32 2 3 2

1 43 2

2). (

2)

22 6 0 ).

7

y xy yx

x

xa x x x b

y xy y x y

+ − =+ + + =

− ++ + = + +

�. ).a �ក�គប"ច�ន�នគ�" x3 3

:3

x

x

++

'ច�ន�នគ�" �

).b �ក�គប"ច�ន�នគ�" 3

3

3:

3

xx

x

++

'ច�ន�នគ�"�

1. �គ !�ប�5k*� ម ABCD � �2�5/ប,- �ជmង ,BC CD ��@ងA� �គ�dប,- ច�ន�ច ,M N

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89ង, !

2

BM CNk

CM DN= = � =ង ,P Q=ម5��ប"'ច�ន�ច�បពB�ប" AM ,

AN នDង BD �

).a ��យបI% ក"J�ក�Cផ7ច��� : PMNQ �n/នDង�ក�Cផ7���� : APQ�

).b គ:6ផ5�ធ@ប�ក�Cផ7 AMN

ABCD

S

S∆

∆ 'Fន�គមនjCន k �

E. ).a ច��Z� 10 , ,

2a b c< < , �ផ7GងH7 �" 2 3 2a b c+ + = �

បfg ញJ 1 2 9

(4 6 3) (3 1) (2 4 154

)a b c b c a c a b+ +

+ − + − + −≥

).b �គ ! , , 0a b c ≠ � �ក��C5�*ចប�ផ���ប"� 2 2 2

2 2 2 2 2 2( ) ( ) ( )

a b cT

a bb c a c ac b+ + + ++

+ += +

K. �គ ! 7Fង̀�"&ន�ប)#ង)#ង'ង 10ន�ងខ3�'ង 130� ��យបI% ក"J �គ)�ង�ក;ន

3Fង̀�" �./ម0� !�គbចផR���P'���� :ម�យ�

N. ក��ងប3ង" Oxy �គ !ប�ច�ន�ច (1;2), (9; 4),BA − ន�ង (5;5)C �

).a �កក*F����នច�ន�ច M �2�5/�ជmង AB ន�ងច�ន�ច N �2�5/�ជmង AC �./ម0� !

||MN BC �O/យ AM CN= �

).b ���ម� �ប67 �" �"=ម (1;2)A �"ប,- កន3�ប67 �" ,Ox Oy ��@ងA� ��ង"

,E F 89ង, !�ប)#ងFង̀�" MN ខ3�ប�ផ�� �

'()&'()&'()&'()&

ចំេល�យ

�. )a 5កqខ:r � 2x ≥ −

3 2 3 3 3( 2) 63 2 3 ( 2) 20 ( 2) 0x x x xx x x x+ −− + ⇔ −= ++ =+

=ង 2t x= + , 5កqខ:r � 0t ≥ 2 2x t⇒ = −

�យ/ង;ន�បព<នQម� �� 2

3 2 3

2 (1)

3 2 0 (2)

x

x

t

xt t

−− + =

=

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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ព� (2) :2

x t

x t

= = −

ច��Z� x t= , ព� (1) �យ/ង;ន� 2 21

20

xxx

x

= − =

− − ⇔

=

ច��Z� 2x t= − , ព� (1) �យ/ង;ន� 2 4 8 02 2 3

2 2 3

x

xxx

= −− −

+=

=⇔

�ផ7GងH7 �"�k/ង# �ញ �យ/ង;ន� 22; 32xx = −= 'U�ប"ម� ��

)b �បព<នQម� �មម*5នDង� 2 2

2 2

1 4 (1)

( ) 2 7 2 (2)

y xy y

y x y x y

x + + + =+ = + +

ព� 2 2 2 2: 1 4 2 8(1) 2 2 2y y xy y y xyx x+ = − − ⇔ + = − −

ជ�ន�ច*5 (2): 2 215 2) 2 (3)(y y y yx xy= − −+

�ប/ 0y = �បព<នQ 3 យ�P'� 2

2

1 0

2 2 0x

x + =+ =

ម�នម�O��ផ5 �6� 0y ≠

(3)មម*5នDង� 2 22( ) 15( ) 05

3x y

x yx y

x y+ + − =

+ = −+ + =

ក�:� ទ��� 5x y+ = − ជ�ន�ច*5 (1) �យ/ង;ន�

2 2( ) 26 ( 4) 01 4 26 0x y y x yxy y y− + = ⇔ ⇔ +− ++ = =+

ម� �An នU�

ក�:� ទ��� 3x y+ = ជ�ន�ច*5 (1) �យ/ង;ន�

2 21 42

( ) 7 0 05

1y

x y yxy y yy

− + = ⇔ −=

+ =+ ⇔

=

Sញ;ន ច��5/យ�បព<នQម� �គM� (1;2) ន�ង ( 2;5)−

�ផ7GងH7 �"�k/ង# �ញ �យ/ង;ន� (1;2) ន�ង ( 2;5)− 'ច��5/យ�ប"�បព<នQ�

�. 3

2 24)

3

33

39

xa A x

x xx

+ − + −= =+ +

��យ x ∈ℤ , �6��./ម0� ! A∈ℤ គM 24

3x +∈ℤ Sញ;ន 3x + '��)ចក�ប" 24

{ }15; 11; 9; 7; 6; 5; 4; 2; 1; 0;1; 3; 5; 92 ; 27; 1x⇒ ∈ − − − − − − − − −− �

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Page 94

)b =ង 3

2 2

3

3

3 3

3

x xB x x C

x x

−= = −+

=+

−+

x∀ ∈ℤ �./ម0� ! B ∈ℤ គM C ∈ℤ� �យ/ង;ន�

2 2

2 2 2 2

3 3 3 3 12. 3

3 9

3 3 3 3

xx x xC x C

x x x x

− ++ +

− = −+

= − −+

=

��យ 2 3), (xCC x∈ ⇒ +ℤ '��)ចក�ប" 12 Sញ;ន { }1; 13; 0; ; 3x ∈ − −

�ផ7GងH7 �"�k/ង# �ញ �យ/ង;ន� ; 10x x= = ន�ង 3x = − �

1.

1. .sin .2)

1 .. .sin2

AMN

APQ

MAN

PAQ

AM ANS AM ANa

S AP AQAP AQ

= =

(11 ). 1AP PM AQ QN PM QN

AP AQ AP AQ

+ + = = + +

=មប�^ប"�យ/ង&ន�

21

1 )1

(BC BM CM k

BM BM k k

+ += = + =

1 ( )2 31DC DN CN CN

kDN DN DN

+= = + = +

=ម�ទD-�បទ=�5 ប*ក��មនDង (2), (3) �យ/ង;ន�

1

PM BM BM k

AP AD BC k= = =

+ ន�ង 1

2 1

QN DN DN

AQ AB DC k= = =

+

ជ�ន�ច*5 (1) �យ/ង;ន�

1 2 1 2( 1)1 1 . 2

1 2 1 1 2 1AMN

APQ

S k k k

S k k k k∆

+ + = + + = = + + + +

� M 2AMN APQS S∆ ∆=

2PMNQ AMN APQ APQ APQS S SS S∆ ∆ ∆ ∆ ∆= − = −

B

A

M C

D

N P

Q

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 95

.*ច�ន� PMNQ APQS S∆=

)b �យ/ង&ន� .

. 1ABM

ABC

S AB BM MB k

S AB BC BC k∆

= = =+

. 1

. 2 1ADN

ADC

S AD DN DN

S AD DC DC k∆

= = =+

�យ/ងកL&ន� 2 11 ;

2

CB DC kk

CM CN k

+= + =

.*ច�6� AMN ABCD ABM ADN CMN

ABCD ABCD

S SS

S S

SS∆ ∆ ∆ ∆− − −=

12 2 2

ADN CMNABM

ABC ACD CBD

S SS

S S S∆ ∆∆

∆ ∆ ∆

= − − −

21 1 2 2

12 1 2 1 ( 1)(2 1) 2( 1)(

2

2 1)

1k k k

k k k k

k

k k

+ + = − + + = + + + + + +

E. )a ច��Z� 0x > � ��យបI% ក"J� 2 1(1

72

2)xx − ≤

2 3 2 31 12

22

7 27x xxx⇔ − ≤ ⇔ +≤

Fន�#�-នj# �មiព Cauchy ច��Z� 3ច�ន�ន� 3 3, ,x x ន�ង 1/ 27�យ/ង;ន�

3 3 3 3 3 231 1 1

227 27

.27

3 .x x xx xx= + + ≥ =+

�យ/ង;ន� 1 2 9

(4 6 3) (3 1) (2 4 1)A

a b c b c a c a b= + +

+ − + − + −

1 2 9

(1 2 ) (1 2 ) (3 6 )a a b b c c= + +

− − − 2 2 2(1 2 ) (1 2 ) (1 2 )

2 3

a b

a b c

a cb c= + +

− − −

Fន�#�-នj# �មiព�ង�5/�យ/ង;ន�

2 2 2(1 2 ) (1 2 ) (1

2 3 2 354

2 ) 1 1 127 27 27

a b c a b cA

a cca bb≥

−+

−+ +

−= + =

មiព�ក/�&ន�ព5� 1/ 3a b c= = =

)b Fន�#�-នj� 2 2 22( )( ) x yx y ≤+ + ន�ង 19 ( , , 0) )

1 1(x y z

x y zx y z

+ + + +

≥ >

�យ/ង;ន� 2 2 2

2 2 2 2 2 2 2 2 22( ) 2( ) 2( )

a b cT

a bb c a c ac b≥

+ + + + + ++ + =

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 96

2 2 22 2 2 2 2 2 2 2 2

1 1 1).

2( ) 2( ) 2 )3

(2( b c

b c a c b aa

a b c+ +

+ + +

= + + − + + +

2 2 2 2 2 2 2 2 222( ) 2( ) 2( )

5b c b a c c a ba + + + + + + + + = ×

2 2 2 2 2 2 2 2 2

1 1 13

2( ) 2( ) 2( )a b c b a c c b a

× + + − + + + + + +

2 3.9 3

5 5T⇒ ≥ − =

.*ច�ន� ��C5�*ចប�ផ���ប" T គM 3

5�

K. �យ/ង��@ប5��ប" ប,- Fង̀�"=ម5��ប"&ន�ប)#ង�ក/ន'5��ប"គM 1 2 7; ; ...;a a a �

�ប/&ន 3Fង̀�" 1 2, ,k k ka a a+ + �ផ7GងH7 �" 1 2k k ka a a+ ++ > �6� 3Fង̀�"�ន�bចផR��;ន'

���� :ម�យ�

zប&ផ7�យមក# �ញ, �យ/ង&ន� 1 2 3a a a+ ≤

42 3

3 4 5

4 5 6

5 6 7

a

a

a

a a

a

a a

a a

a a

+ ≤+ ≤+ ≤+ ≤

=មប�^ប"� 1 2 3 4 5 6 7, 10 20, 30, 50, 80, 130a a a aa aa> ⇒ > > > > >

# �មiពច�ង�� យផ7�យនDងប�^ប"ព� ប,- �ប)#ងFង̀�"�*ច'ង 130�

.*ច�ន� &ន 3Fង̀�").5bចផR��;ន'���� :�

N. )a ��យ ||MN BC �6� AM ANk

AB AC= =

�យ/ង&ន� (1)(0 1)

(2)

AM k ABk

AN k AC

=≤

=

� �

� �

)���យ� (8; 6); (4;3)AB AC= − =� �

ព� 2 2(1) 8: ( 0) 16A k kM AB k= = + − =

ព� (2) : AN k AC=� �

(1 )NC AC AN AC k AC k AC= − = − = −� � � � � �

2 2(1 ) 4 5(1 )3NC k k⇒ += − = −

C B

A

M N

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 97

110 5(1 )

3AM NC k k k= = − =⇔ ⇔

) 8

3( ) 6 2 2 0

3( )

8 1113( 3

4 4 711

3 33 3( ) 3

31

3

1 2 3

MM A

M A M

N AN

N A

N

xAM AB y y

x xx

x

AN ACy

x

y

yy

=− == − = − = − + =

+ =

⇔− = = + ==− =

= + =

� �

� �

.*ច�ន� 11 7; ; ;

3 30 3M N

)b zប&J ប67 �" d ).5��e#�ក �"កន3�ប67 �" ,Ox Oy ��@ងA� ��ង" ( ;0)M m ន�ង (0; )N n

ច��Z� ( , 0)m n > �

ម� � ( )d &ន^ង� 1x y

m n+ =

��យ (1;2)A d∈ �6��យ/ង;ន� 1 2 2 1 21 1

1

m

m nn

n m m⇔ ⇔ =+ = = −

, 00m n> > .*ច�6� 1m >

�យ/ង&ន ( ; )MN m n= −�

, �6� 2 2

2 2 2 2 22 22

1 1

mm mMN

m mn m = + = + = + + − −

=ង 1t m= − Sញ;ន 1m t= + , �ព5�6�

2

2 2 22

2 8 42( 1) 2 1 4t

ttM

t ttN + +

= + + = + + + +

2 332

4 4 45 3 16 3 4 5t t t

t t t = + + + + +

+

+ +

2

2 3 33

2

3 16 3 4 5

4

min4

4t

tMN t

tt

= = =

= + + ⇔ ⇔

.*ច�ន� ម� �ប67 �" d គM� 3

3 3

3 31

4.4 2( 4 1) 0

4 1 4 12

2 )(

xx

yy+ = ⇔ + − + =

+ + �

'()&

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វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី១៩

�. �គ !���� : ABC � =ង D 'ច�ន�ច�2�5/ BC � �2�5/�ជmង AB ន�ង AC �គ�d

ប,- ច�ន�ច P ន�ង Q � ប,- ប67 �" �"=ម P ន�ង Q �បនDង AD=ម5��ប" �"

ប,- �ជmង BC ��ង" N ន�ង M � ��យបI% ក"J �ក�Cផ7 ( MNPQ ) max{≤ �ក�Cផ7

( ),ABD �ក�Cផ7 ( )}ACD � មiព�ក/�&ន�2�ព5,?

�. zប&J , ,a b c 'ប�ច�ន�ន# �ជ%&ន�ផ7GងH7 �"5កq:r 1abc = �

��យបI% ក"J� 3 3 3

2 23 (*)

( ) ( )

2

( )b c c aa b ac b≥

+ ++

++ �

1. �គ ! 4 3 2( ) axf x cx bx x d+ + + += , ច��Z� , , ,a b c d 'ប,- ច�ន�ន�ថ�� zប&J

(2) 2(1) 0, (3) 3010, f ff = = = � ច*�គ:6 (12) ( 8)15

10

f f+ − + �

E. �គ !ច��� :�;9ង ABCD � �2�5/ប,- Fង̀�" , , ,AB BC CD DA �គ�dប,- ច�ន�ច

, , ,M N P Q 89ង, ! AQ DP CN BM= = = � ��យបI% ក"J �ប/ MNPQ ' ��

�6� ABCD ' ���

K. �����យន�ងព�iកyម� �� 3 2 4 4)( (4 14 )x x x x a x− − − + − = ≥− �

N. �����យម� ��ង�� ម� 2 21 1

21

2

x xa a

a a

+ −

− = , ច��Z� 0 1a< < �

'()&'()&'()&'()&

ចំេល�យ

�. =ង APx

AB= ន�ង AQ

yAC

= , ច��Z� 0 , 1x y< <

�យ/ង;ន ( )

( )

.

.APQ

ABC

S AP AQxy

S AB AC= =

Sញ;ន ( ) (1){( )} . ABCS APQ xy S=

មu9ង�ទ@�, �យ/ង&ន� B

A

C N D M

P

Q

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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1BN BP

xBD BA

= = − ន�ង 1CM CQ

yCD CA

= = −

Sញ;ន 2( ) ( )(1 ) (2)BNP ABDx SS = −

2( ) ( )(1 ) (3)CMQ ACDy SS = −

ព� (1), (2) ន�ង (3)�យ/ង;ន�

( ) ( ) ( ) ( ) ( )MNPQ ABC APQ BNP CMQS S S SS = − − −

2 2( ) ( )(1 ) (1 ) . (1 ) (1 ) .ABD ACDxy x S xy y S+ = − − − − − −

2 2( ) ( )2 . 2 .ABD ACDx xy x S y xy y S+ = − − − −

��យ 2 2(2 ) 0, 2 (2 ) 02 x y x y xy y y xx y x yx = − − > − − = − −− >−

�6� { }2 2( ) ( ) ( )(2 .m x) a2 ;MNPQ ABD ACDyS x xy Sx yy x S≤ + − − − −

{ }2( ) ( ) ( )2( ) ( ) .m ;axMNPQ ABD ACDS x y x S Sy ⇔ + − +≤

{ }2( ) ( ) ( );1 ( 1) .maxMNPQ ABD ACDS x y S S − + − ⇔ ≤

.*ច�6� { }( ) ( ) ( )max ;MNPQ ABD ACDS SS ≤

Iw មiព�ក/�&ន5���=)� ( ) ( )ABD ACDS S= ន�ង 1,x y+ =

� M BD DC= ន�ង 1AP AQ

AB AC+ = �

�. 3 3 3

3( ) (

2

)

2

) (

2

b c c a a ba b c≥

+ + ++ +

3 3 3

1 1(1)

( ) ( 2)

1

(

3

)b c c aa c ab b⇔ ≥

+ ++

++

=ង 1 1 1; ;x

a b cy z= = = �ព5�6� 1xyz = (2)

�O/យ 2 2 2

(1) )3

3(2

x y z

y z x z x y+ +

+ +≥

+⇔

Fន�#�-នj# �មiព Cauchy �យ/ង;ន�

2 2 2

(4), (5), (6)4 4 4

x y z y x z z y x

y z xy

zz

xx

y

+ +≥ ≥ ≥++ + ++ + +

ប*កFងRនDងFងRCន# �មiព (4), (5) ន�ង (6) �យ/ង;ន (7)

��]ព��6� 33 3

2 2 2(**)

xyzx y z =≥+ +

Page 101: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 100

ព� (**) ន�ង (7) �យ/ងSញ;ន# �មiព (*) �

1. =ង ( ) ( ) 10g x f x x= − , �ព5�6� (1) (2) (3) 0g g g= = = �6� ( )g x )ចក�ច"នDង

( 1)( 2)( 3)x x x− − − �

��យ ( )g x 'ពO�o.M��កទ� 4 �6� 0( ) ( 1)( 2) ) )( 3 (g x x x x x x= − − − −

0( ) ( 1)( 2)( 3)( ) 10f x x x x x x x⇒ = − − − − +

�ព5�6� (12) ( 8)15 1984 15 1999

10

f f+ − + = + =

E. =ង �,, ;,A AB a MNQ DP CN BM x AM au PQx D α= = = = = −= = =

ព�ន��!���� : AMQ �យ/ង&ន�

2 2 2( ) 2 ( )cos (1)a x x x a x Au = − + − − .

�ព5 MNPQ ' ��គM � 090AQM α= −

ន�ង MN NP PQ QM u= = = = �

�យ/ង;ន �2 2 2 2 . .cosAQ QM AQ QM QMAM A= + −

2 2 2 2 si( ) nu xa x ux α⇔ = + −− (2)

Fន�#�-នj�ទD-�បទFន�គមនj��ន�ក��ង���� : QDP �យ/ង;ន�

sinsin s

nin

six u

uD

x Dαα

⇒ ==

ទ�6ក"ទ�នង (2) 3 យ�P' 2 2 2 22 ( )i 3( ) s nu x xa x D− = + −

ព� (1)ន�ង (3)Sញ;ន cos (1 sin ) 0x

A Da x

= − ≥−

��យច��� : ABCD �;9 ង �6� 00 90A< ≤

��យបI% ក".*ចA� ).� �យ/ង;ន 00 90B< ≤ ន�ង 0 00 ;90 900C D< <≤ ≤

Sញ;ន 0360A B C D ≤+ + +

.*ច�ន� �យ/ង��e#&ន 090A B C D= = = =

.*ច�ន� ABCD ' ���

K. .�ប*ង�យ/ង&ន5កqខ:r 4x ≥ � ម� � (1)bច����k/ង# �ញ'�

( ) ( )2 2

4 1 4 2 4 1 4 2x x a x x a− − + − − = − − + − − =⇔

B

A

C

D

Q

P

M

N

x

x

x

x

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 101

=ង (4 0)y x y= ≥− , ម� � 3 យ�P' 1 2 (2)y y a− + − =

5កqខ:r �./ម0� ! (2)&នUគM 0a > , ��Z��ប/ 0a < គM�ឃ/ញJម� �An នU�

�ប/ 0a = �6� 4 1x − = ន�ង 4 2x − = (ម�នម�O��ផ5)

• ក�:� �� �ប/ 0 1y≤ < �6� 31 2

2

ay y a y= ⇔ −− + − =

�យ/ង��e#&ន� 33

0 1 0 2 312

aa a≤ ⇔ ≤ − < ⇔− < ≤<

.*ច�ន� ច��Z� 1 3a< ≤ �6� [ )3

;12

0a

y ∈−=

�ព5 1a ≤ � M 3a > �6�ម� �An នU

• ក�:� �� �ប/ 1 2y≤ ≤ �6� 2 11y y a a+ − = ⇔ =−

.*ច�ន� ច��Z� 1a = �6�ម� � (2)&នU :y 1 2y≤ ≤

ច��Z� 1a ≠ �6�ម� � (2) An នU

• ក�:� 1� �ប/ 2y > �6� 31 2

2

ay y a y= ⇔ +− + − =

�យ/ង��e#&ន 32 1

2a

a + > ⇔ >

.*ច�ន� ច��Z� 1a > �6�ម� � (2)&នU 3

2

ay

+=

�ព5 1a < គMម� � (2)An នU

��បមក� ច��Z� 1a = គMម� � (2)&នU :y

2 2 51 81 4y xx −≤ ≤ ⇔ ≤ ≤ ⇔ ≤ ≤ .

�ប/ 1 3a< ≤ ម� �&នU�

2 22

2 22

3 34

2 2

3

4

34 4

22

ax

a ax y x

ax y

− − − = =

= +⇔

+ = = +

+ − =

�ប/ 1a < ម� �An នU

�ប/ 3a > គMម� �&នU 2 6 2

4

5aax

+ +=

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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N. 22 2 211 1

11

2 21

2 2

xx x x

aa a a

a a a a− = ⇔

+ − −

+

=

,

)ចកFងRS�ងព���ប"ម� �នDង 21

2

xa

a

+

�យ/ង;ន 2 2

2 11

1 1

x xa a

a a+ − = + +

��យ (0;1)a ∈ �6�&ន 0;2

πϕ ∈

�./ម0� ! tan2

aϕ =

.*ច�6� ម� �bច����k/ង# �ញ;ន'�

( ) ( )2

2 2

2 tan 1 tan2 21 sin cos

1 tan 1 tan2

1

2

x x

x x

ϕ ϕ

ϕ ϕϕ ϕ

− = + +

= +

+

Fន�គមនj ( ) ( )( ) sin cosx x

f x ϕ ϕ+= 'Fន�គមនjច��'ន�ចa �O/យ&ន (2) 1f = �

.*ច�ន� 2x = 'U)�ម�យគ�"�ប"ម� ��

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២០

�. �����យម� ��ង�� ម� 22. 2 5 27 144 191x x x− +− =

�. �គ !ច�ន�នគ�"# �ជ%&ន n ន�ង 1 2 3 4d dd d< < < ' 4��)ចកគ�"# �ជ%&ន�*ចប�ផ���ប" n �

�ក�គប"ប,- ច�ន�នគ�"# �ជ%&ន n �./ម0� ! 2 2 2 21 2 3 4n d d d d+ + += �

1. �2�5/�ងBង"ម�យ�គ !�;�ច�ន�ច� =មទ��បជ��ទ�ងន"�ប"ប�ច�ន�ច �គង"ប67 �")កងនDង

ប67 �" �"=មព��ច�ន�ច�ទ@�� ��យបI% ក"J 10ប67 �").5ទទ�5;ន �"A� ��ង"

ច�ន�ចម�យ�

E. �គ ! , , 0a b c > ន�ង 1a b c+ + = � បfg ញJ� 2 2 2

30 (11 1 1 1

)ab c b bc caa

+ + ≥++ +

K. �គ !���� : ABC &នម��S�ងប�'ម���|ច, ��យបI% ក"J� 1 1 1 3

1 tan tan 1 tan tan 1 tan tan 1 2 3A B B C C A≤+ +

+ + + + + + +

N. �គ !�ងBង"ផa�� O ��n/ a , &នFង̀�"ផa��ព��)កងA� គM AB ន�ង CD � �2�5/កន3�ប67 �"

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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CD �គ�dព��ច�ន�ច ,M N 89ង, ! CN OM=� �

� ប67 �" AM �"�ងBង"��ង" P �

ច*�ព�ន��!�ម/5�ព5 N )�ប�ប|5�2�5/Fង̀�" CO , ���� : ANP &ន)កង��ង" N � M

�ទ? �ប/���� : ANP )កង ��/�ព5�6� N 4���2ទ�=�ង,? '()&'()&'()&'()&

ចំេល�យ �. ម� � 2 48 64) 1 2.3.(9 2 5x xx⇔ − + − −=

23.(3 8) 2 51 2.x x⇔ − =− −

2

23. (3 8) 16

2

52 5x

x − −

=

+⇔ −

5កqខ:r � 2

24

144 19

3

9272 3

1 04

9

xx x

x

+≥− + ≥ ⇔

−≤

=ង 2(3 8)

2

5xy

−= + �យ/ង;ន�បព<នQម� �� 2

2

2 5

(3 8) 2 5

(3 8) x

x

y

y

= −− = −

2 2 2

2 2

48 64 2 5 ) 48.( ) 2( )

9 48 64 2 5 9 48 64 2 5

9 9.(y x x y x x y

x

y

x y

y

y y x

− + = − − − − = −⇔ ⇔

− + = − − + =

2

( ).(9 9 46) 0

489 64 2 5y

y x y x

y x

− +⇔

− + = −

− =

2

(1)48 64 2 5

0

9 xy y

y x⇔

− + − =

− = � M 2

(2)48 64

9 9 46

2

0

9 5y x

y x

y

+ − =

+ − = −

ក�:� ទ���

2 2

3(1) 23

48 64 2 5 50 69

09 99

x yy x y x

x yy yy x y⇔ ⇔ ⇔

− + = − −

= == = = = + =

ក�:� ទ���

2

2

46 99 9 46 0 23

48 64 2 541

(2) 99

59

81 4 29 0

x y

yy x x

y y xyy

−+ − = =

⇔ ⇔ ⇔ = =− + = − − + =

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 104

.*ច�ន� { }3S = '�ន��U�ប"ម� ��

�. �ឃ/ញJ 2 0 (mod 4)x ≡ �ព5 x គ*, 2 (m1 4)odx ≡ �ព5 x ��

�ប/ n 'ច�ន�ន� �6��គប"ប,- ច�ន�ន id �ទQ)���O/យ

2 2 2 21 2 3 4 1 (mod1 1 1 0 4)dn dd d≡ + + + ≡ + + + ≡ (ក�:� �ន�ផ7�យព� �ព��)

.*ច�ន� �យ/ង;ន 2n k=

�ប/ 4'��)ចក�ប" n �6� 1 1d = ន�ង 2 22 3 41 02, nd d d+≡ ++= )ចកម�ន�ច"នDង 4 (ក�:�

�ន�ផ7�យព� �ព��)� .*ច�ន� �យ/ង;ន n )ចកម�ន�ច"នDង 4�

.*ច�ន� { } { }1 2 3 4, , , 21 ,, ,d d d p qd =

� M { } { }1 2 3 4 1,, , , 2, , 2d pd d d p= ច��Z� ,p q 'ប,- ច�ន�នប[ម�

ក��ងក�:� { } { }1 2 3 4, , , 21 ,, ,d d d p qd = �យ/ង;ន (m3 4)odn ≡ (ផ7�យព� �ព��)

.*ច�ន� ( )25 1n p= + �O/យ n )ចក�ច"នDង 5 , �6� 3 5p d= = ន�ង 130n = �

1. =ង 1G 'ទ��បជ��ទ�ងន"�ប" 3ច�ន�ច 1 2 3, ,A A A �

ង" 1 1 4 5KG A A⊥ ន�ង 1 4 5ON A A⊥ �

�ព5�6�, �យ/ង;ន� ( )1 1 2 3

1

3OG OA OA OA= + +� � � �

=ង 1M 'ច�ន�ច,ម�យ �ប"ប67 �" 1 1G K , ម� �#� �ចទ<�

ច��Z�ប67 �")កង 1 1G K គM�

( ) ( )1 1 1 1 1 2 3 1 4 5 1

1 1,

3 2OM OG ON OA OA OA OA OAα α α= + = + + ++ ∈� � � � � � � �

.*ចA� ).�, �យ/ងទទ�5;នប,- ម� ��ង�� មច��Z� 9ប67 �"�ផpង�ទ@��

( ) ( )2 2 3 4 2 1 5

1 1

3 2OM OA OA OA OA OAα= + + + +� � � � � �

..............

( ) ( )10 3 4 5 10 1 2

1 1

3 2OM OA OA OA OA OAα= + + + +� � � � � �

�ប/�យ/ងយក 21 10...2

3α α α= = = = �6� 1 2 10...OM OM OM OM= = = =

� � � �

Sញ;ន, 10ប67 �"�ង�5/ �"A� ��ង" M , �2��ង"�6� ( )1 2 5

1...

3OM OA OA OA= + + +� � � �

1A

2A

3A

4A

5A

1N

1M

1K

O

1G

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Page 105

E. Fន�#�-នj# �មiព Cauchy �យ/ង;ន�

( ) 1 19

1ab bc ca

ab bc ca + + + +

Iw �n/�ក/�&ន�ព5 ab bc ca a b c= = ⇔ = =

�យ/ង;ន� 2 2 2 2 2 2

1 1 1 1 1 9

a ab bc cab c ba ab bc cac+ + +

++

++ + +≥

+

( )( )22 2 23

(3 7

2)ab bc caa ab bc cb ac

++ ++ +

≥+ +

Iw �n/�ក/�&ន�ព5 2 2 2b c ab bc c ba a a c+ + = + + ⇔ = =

មu9ង�ទ@�� ( )( )2 2 2

2 2 232( )

3a ab bc ca

a b c ab bc cab c

+ + + + +≤+ ++ +

2( )

3(

1

33)

a b c+ += =

�O/យ 2() 1 (4)( )3 aab bc c b ca ≤ + + =+ + , Iw �n/�ក/�&ន�ព5 a b c= = �

ព� (3)(2), , (4)�យ/ងSញ;ន� 2 2 2

1 1 19 21

130

a b ab bcc ca+ + + ≥ + =

+ +

Iw �n/�ក/�&ន�ព5 1

1 3

a b c

a ba

cb c

= =⇔ = = =

=+ +

K. =ង tan , tana , ,t n y B zx A C== = �យ/ង;ន x y z xyz+ + =

�យ/ង;ន 1 1 1

1 tan tan 1 tan tan 1 tan tanA B B C C A+ +

+ + + + + +

1 1 11 1 1

x y z

x y y z z x x xy xz y yz yx z zx zy= + + = + +

+ + + + + + + + + + + +

zប&J A B C≥ ≥ �6� x y z≥ ≥ , ព��6�, �យ/ងSញ;ន�

y yz yx z zx zy

x y z

x xy xz y y

x x

z yx z

y z

x y

x

z z

≥ + + ≥ + +

≥ ≥+ + + +

+ +

+ +

Fន�#�-នj# �មiព Bunyakovski , �យ/ង;ន�

.( ) 3( )x y z

x xy xz y yx yz z zx zyx xy x

x yz y yx yz z zx zy

z

+ + + + + + + + + + + + + + + ++

≤ +

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 106

3( ) 32( )2( ) 1

x y z x y zxy yz zxx xy xz y yx yz z zx zy x y z xy yz zxx y z

+ ++ + = + ++ + + + + + + + + + + ++

+

3 3

3 1 2 31 2

xyz

x y z

=++

+ +

Iw " "= �ក/�&ន�ព5 3

x y z A B Cπ⇔ = = == = �

N. ��ជ/�� /����យក*F����ន�. �)កង Oxy 89ង, ! O ��|�A� នDងផa�� O �ប"�ងBង",

F<កp Ox ��|�A� នDងកន3�ប67 �" OB ,

F<កp Oy ��|�A� នDងកន3�ប67 �" OC �

=ង CN OM l= = ច��Z� 0 l a≤ ≤

�យ/ង;ន ក*F����នប,- ច�ន�ច�

0), ( , 0), (0, ), ( , 0), (0, )( , 0, ( ,0 )B a C a A a N a l M lO − − − �

�មគ�:�;ប"ទ� m �ប"ប67 �" AN គM N A

N A

y

x

y a lm

x a

− −= =−

ម� � (1:

)0

x y a y lAM lx ay xal

a l l⇔ ⇔ = − ++ = + + =

− −

ក*F����នច�ន�ច�បពB�ប" AM ន�ង�ងBង"ផa�� O គM'U�ប"�បព<នQ�

2 2 2

(1)

(2)

0

x y a

lx ay al+ =

+

+

=

ព� (1)ន�ង (2)�យ/ង;ន� 2 2

2 22

( )ay

l

y la++ =

( ) ( )2 2 2 2 2 2 222 0 0a yl y a y lal yl a + =⇔ + = ⇔ + +

2

2 2

0

2 l

l

y

ay

a

+

= = −

ច��Z� 0y = , �យ/ង;ន x a= − , �ន�'ក*F����នច�ន�ច ( ;0)A a−

ច��Z� ( )2 22

2 2 2 2

2P P

llxy

a l l

a aa

a

−= − ⇒ =

+ +

B A

C

N

M

O

P

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 107

�មគ�:�;ប"ទ� k �ប"ប67 �" :NP ( ) ( )( )

2 2 2

2 2

2N P

N P

a l a l ly ayk

x ax l a

+− − +=

− −=

���� : ANP )កង��ង" N AN NP⇔ ⊥ 11 kkm

m⇔ ⇔ = −= −

.*ច�ន� ( ) ( )( )

2 2 2

2 2

2

1

l la l a a aAN

aaNP

al

− +=

+⇔

− −⊥

( ) 2 00

la l l

l a

=− =

= ⇔

.*ច�6�, ���� : ANP )កង��ង" N �ព5 N C≡ � M N O≡ �

ប,- ក�:� �ផpង�ទ@� ���� : ANP ម�ន)កង��ង" N �ទ�

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២១

�. �����យ�បព<នQម� ��ង�� ម� 2 2

5 3 5 3

4 1

16 20 5 512 160 10 2 0

y

x x x y y y

x + =

− +

+ − + + =

�. �ក�បព<នQគ*ច�ន�នគ�" ( ; )m n �./ម0� ! 2 2p nm= + 'ច�ន�នប[ម �O/យ 3 3 4m n+ − )ចក

�ច"នDង p �

1. �គ !���� : ABC &នម���|ច 3ន�ងប,- �ជmង���� :�ផ7GងH7 �"5កq:r

AC AB BC< < � �2�5/ប,- �ជmង AB ន�ង BC �គ�dប,- ច�ន�ច��@ងA� K ន�ង M

89ង, ! AK CM AC= = � =ង O ន�ង I =ម5��ប"'ផa���ងBង"\� Dក��] ន�ង

\� Dកក��ង���� : ;ABC R′ ' ��ងBង"\� Dក��]���� : BMK �

��យបI% ក"J ' , KR OI M OI= ⊥ �

E. �គ ! , , ,a b c d 'ប,- ច�ន�នព��# �ជ%&ន�ផ7GងH7 �"5កq:r � 1 1 1 14

a b c d+ + + = �

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 108

បfg ញJ� 3 3 3 3 3 3 3 3

3 3 3 3 2( ) 4 (2 2 2

1)2

b c d aa b db d

ca c+ + ++ + + + ≤ + + + −

K. �ព5បងB�5�ក� 0180 �2�5/ប3ង" �6�ប,- �5ខ 0;1; 8គMម�ន)�ប�ប|5, ប,- �5ខ

6ន�ង 9ប-*�ក)ន3ង !A� , �ប,- �5ខ�ផpង�ទ@�ម�ន&នន<យ�ទ@��ទ� ��/&នច�ន�ន

).5&ន�5ខ 7ខ7ង"ច�ន�នប9�6n ន ).5�ព5�គបងB�5�ក��P 0180 នDងម�ន&ន �

)�ប�ប|5? ផ5ប*ក�គប"ប,- ច�ន�ន�6��n/នDងប9�6n ន?

N. �គ ! O 'ច�ន�ចនDងម�យ�2�5/ប67 �" d នDងម�យ� P ន�ង Q =ម5��ប"'ប,- ច�ន�ច

ច5<��2�5/�ងBង"ផa�� O � a ន�ង�ងBង"ផa�� O � b , 89ង, ! d 'ប67 �"ព��ក��ង

�ប"ម�� POQ , a ន�ង b '�ប)#ងព��ម�ន)�ប�ប|5 a b> � ព�ន��!ច�ន�ច M 89ង, !

OM OP OQ= +� � �

� �ក�ន��ច�ន�ច ច�ន�ច M � '()&'()&'()&'()&

ចំេល�យ

�. =ង 2t y= , ច*5ក��ង�បព<នQ �យ/ង;ន�

( ) ( ) ( )

2 2

5 5 3 3

1 (1)

16 20 5 2 (2)

t

x t x t x t

x + =

+ −

+ + + = −

=ង [ ]0;2cos

sin,

t

x αα π

α∈

=

= � ជ�ន�ច*5ម� � (2) �យ/ង;ន�

( ) ( )5 3 5 316sin 162 c0sin os5sin 20cos 5cos 2α α α α α α+− − + = −+

sin 5 sin 1cos5 24

πα α α = −

⇔ + = − ⇔ +

0

( )3 2

2 5k k

π πα⇔ = +− ∈ℤ

��យ [ ]0;2α π∈ �6�ព� (3)�យ/ង;ន�ក;ន� 13 21 29 37; ; ; ;

4 20 20 20 20

π π π π πα =

.*ច�ន� �បព<នQ).5 !&នប,- ច��5/យ� 2 2 13 1 13; , sin ; cos ,

2 4 20 2 20

π π

21 1 21 29 1 29 37 1 37sin ; cos , sin ; cos , sin ; cos

20 2 20 20 2 20 20 2 20

π π π π π π

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 109

�. �យ/ង&ន� 3 3 0 (mo4 )d ( ) 4 0 (mo ) (2)dm mn pnn p m+ − ≡ ⇔ − + − ≡

3 ( ) 12 (mod0 )pmn m n⇔ ≡+ + .

ប*ក��មនDង� 3 3 (mod4 0 )m n p+ − ≡

2 2 2( 2)( 0 (mo2( ) 4) )dn nm m n pm n m⇔ + + − + + ≡+ + .

��យ p 'ច�ន�នប[ម�6��យ/ង&ន 2 5ទQiព

� ក�:� ទ��� �ប/ 2 22 ( )m n m n+ ++ ≡

Sញ;ន� 2 2 1( 1) ( 2)

12 1

mm m m n n

nn m n

=− ++ ≤ + + ⇔ ≤−

=⇔

� M 2

1

m

n

= =

� M 1

2

m

n

= =

�កជ�ន�ច*5 �យ/ង�ឃ/ញJ� �បព<នQច�ន�ន ( ; )m n គM (1;1),(2;1)ន�ង (1;2)�ផ7GងH7 �"�បoន.

� ក�:� ទ��� �ប/ 2 2 2 22 2( ) )(4n mn m nm nm+ + − + + +⋮

2 22 2( ) 4 ( )nmn m n m− +⇒ ++ ⋮ .

��យ 2 2( ) 4 ( 1)( 1) 1 0mn m n m n− + + = − − + >

Sញ;ន 2 22 2( ) 4 mm n nn m ≥+ +− + (�ក/�&ន)��ព5 1m n= = )

ន���h ន� �គប"ប,- គ*ច�ន�ន ( , )m n �ផ7GងH7 �"�បoនគM� (1;1),(1;2)ន�ង (2;1)�

1. =ង ,N P ��@ងA� 'ច�ន�ចក,- 5�ប"�ជmង ,AB BC � ប,- ច�ន�ច ,L Q 'ច�ន�ចប9�

�ប"�ងBង"\� Dកក��ង �PនDង ,AB BC � �យ/ង=ង , ,Aa C A cB bC B= == �

=មប�^ប"�យ/ង;ន , BBK Mc b a b= − = − (��Z� b c a< < )�

មu9ង�ទ@�, �យ/ង&ន�

2

a bLN BL BN

−= − = ,

2

c bPQ BP BM

−= − =

�2�5/ IQ ន�ង IL �យ/ង�dប,- ច�ន�ច��@ងA� គM

S ន�ង H 89ង, ! || , ||OS BC OH AB , �ព5�6�

,2 2

OHc b a b

O NS PQ L= =− −= =

B

A

C Q P M

K

L

N

I O

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 110

���� :S�ងព�� OSH ន�ង BKM &ន^ង.*ចA� �O/យផ5�ធ@ប.�ន*ច 1

2

OH

BMλ = = �

មu9ង�ទ@� ��យ���� :S�ងព��.*ចA� �6�ផ5�ធ@ប.�ន*ច �n/នDងផ5�ធ@ប �Cនប,-

�ងBង"\� Dក��]�ប"���� :S�ងព���6��

ច��� : OHIS \� Dកក��ង�ងBង"Fង̀�"ផa�� OI , .*ច�6� ��ងBង"\� Dក��]���� : OSH

�n/ 2

OI � ព�ប,- 5ទQផ5�6� �យ/ងSញ;ន 'R OI= �

ចt"," ប,- កន3�ប67 �" OH ន�ង OS &នទ��d.*ចA� នDងកន3�ប67 �").5��e#A�

គM BA ន�ង BC �6�=មចtប"� �ក�5=ម OB�

ប�)5ង H �P' 'H 4���2�5/កន3�ប67 �"

,BA S �P' 'S 4���2�5/កន3�ប67 �" BC �O/យ I �P' 'I � ចtប"ឆ3���ធ@បនDង ប67 �"

ព���ប" BAC∠ ប�)5ង 1 1,' 'H H S S֏ ֏ ន�ង 1 1,'B BH B BH S S′ == �

�យ/ង;ន 1 1

2BM BK

BH BS= = , .*ច�6� 1 1 ||S H MK � ចtប"ប�)5ង\�ងផa�� B ប�)5ង 1H M֏

�6� 1 1 2,S K I I֏ ֏ 4���2�5/Fង̀�"ផa���ប"�ងBង"\� Dក��]���� : BKM គ*ព� B

ព��6� Sញ;ន KM OI⊥ �

E. + .�ប*ង�យ/ង��យបI% ក"� 3 3 2 2

3

2(*)

2

b a ba + +≤ ច��Z��គប" 0,a b >

ព��'.*ច�ន� # �មiព 4 2 2( )( ( 0*) ) a ab ba b + +−⇔ ≥ � Iw " "= �ព5 a b= �

+ Fន�#�-នj# �មiព (*) , �យ/ង;ន�

3 3 3 3 3 3 3 3 2 2 2 2 2 2 2

3 3 3

2

3

2 2 2 2

a b c d a b c d

a b b c c d d

b c d a b c d a

a+ + + + + +

++ + + + + + + +≤

+ + + (1)

+ �./ម0���យបI% ក" (1) , �យ/ង��e#��យបI% ក"�

2 2 2 2 2 2 2 2

2( ) 4 (2)a b c d

a b b c c d

b c d aa b c

dd

a

+ + + + ≤ + + + −+ + ++ + + +

�យ/ង&ន�

2 2 2 2 2 2 2 2

(2 4)b c d da b c a

a b b c c d a da b b c c d a d

+ − + + − + + − + + − + + + +

+ +

+ +

⇔ ≥

2 2 2 24

ab bc cd da

a b b c c d d a+ +⇔ +

+ + +≥

+1 1 1 1

1 1 1 1 1 1 12

1a b b c c d d a

+ + ++ + + +

⇔ ≥ (3)

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 111

+ មu9ង�ទ@�� 1 1 1 1 162

1 1 1 1 1 1 1 1 1 1 1 12

a b b c c d d a a b c d

+ + + = + + + + + + +

Iw " "= �ក/�&ន�ព5 a b c d= = = � Sញ;ន (3)ព��

.*ច�ន� # �មiព (1) ��e#;ន��យបI% ក"� Iw " "= �ព5 1a b c d= = = = �

K. =ង ច�ន�ន).5��e#�ក��យ ( )1 2 7 1... 0a a a a ≠ � ច��Z� 1a &ន 4 ��ប@ប�� /គM 1;8; 6;

ន�ង 9, ច��Z� 2a ន�ង 3a &ន 5��ប@ប�� / ()ថម�5ខ 0 ), ច��Z� 4a &ន 3��ប@ប�� /

(��Z�'ច�ន�ន�2ក,- 5 �6���e#�\5 6ន�ង 9)� ប,- �5ខ�25"� 5 6 7; ;a a a

ចt","J?��e#�ផ7GងH7 �"�A5 �:j " ឆ3��ផa��" , 'ក")-ង 5a �� យ�ព5បងB�5

�ក� ��e#;នប�)5ង' 3 6,a a ប�)5ង' 2 7,a a ប�)5ង' 1a � .*ច�6� ប,- �5ខ

5 6 7; ;a a a ��e#;នក�:�"បI% ��យប,- �5ខ 3 2 1; ;a a a =ម5��ប" �O/យច��Z���ប@ប

�� / 1 2 3; ;a a a &ន)�ម�យ��ប@បគ�" ក�:�" 7 6 5; ;a a a =ម5��ប"�

.*ច�ន� ច�ន�នប,- ច�ន�ន).5�ផ7GងH7 �"�បoនគM� 4.5.5.3 300= ច�ន�ន�

�យ/ង�ឃ/ញJ �5ខន�ម�យ� 1;8; 6; 9&នម�ខម-ង.*ចA� �2ទ�=�ង 1a ក��ងប,- ច�ន�ន�ង

�5/� .*ចA� �ន�).� ច��Z�ប,- ទ�=�ង�ផpង�ទ@��

.*ច�6� ផ5ប*ក�គប"ប,- ច�ន�ន�6��n/�

( ) ( ) ( ) ( )6 5 4 2300 3001 8 6 9 . . 10 0 1 8 6 9 . 10

41 10 10

510+ ++ + + + + + + + ++

( ) 33000 1 8 . .10 1959460

3200+ + + = �

N. ប�ង̀/�����យក*F����ន)កង &នគ5"����យគM O , F<កpbប"�� Ox &នផ7�កប67 �" d , F<កp

F����ន Oy )កងនDង Ox ��ង" O �

ប,- ច�ន�ច , ,M P Q =ម5��ប"&ន

ប,- ក*F����នគM ),( ; ( ; ),P Px xy y

( ; )Q Qx y � ប,- ក*F����ន�ន� =ម

5��ប"កL'ក*F����ន�ប"ប,-

#� �ចទ<� , ,OM OP OQ� � �

x

y

d O

Q

P

M

Page 113: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 112

=មប�^ប"� OM OP OQ= +� � �

.

�យ/ង;ន ;P Qx x x= + ន�ង P Qy yy= +

��យ Ox 'ប67 �"ព��ក��ង�ប"ម�� POQ �6��ប/ OP &នម� � y kx= គM�យ/ង;ន

OQ &នម� � y kx= − �

��យ 2 2 2P Px y a+ = �6�

22 2 2 2 2

21P P P

ax xk ax

k=+

+⇔ =

ព��6� 2 2

221P

aky

k+=

.*ចA� ).� �យ/ង;ន� 2 2 2

2 22 2

;1 1Q Q

by

b kx

k k+ += =

យកច��-ទ�ក�ក"J d 'ប67 �"ព��ក��ង�ប" �POQ �6� . 0, . 0P Q P Qx yx y> < , .*ច�6�,

ព� P Qx xx= + �យ/ង;ន� 2

2 2 22

( )

12 (1)P Q P Qx x x x

a bx

k= + +

+= +

ព� P Qy yy= + �យ/ង;ន� 2 2

2 2 22

( )

12 (2)P Q P Q

ky y y y

a by

k

−=+

= + +

ព� (1) ន�ង (2) Sញ;ន 2 2

2 21

( ) ( )

x y

a b a b+ =

+ − (3)

.*ច�ន� �ន��ច�ន�ច�ប" M '�F5�ប&នម� � (3) �

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២២

�. �����យ�បព<នQម� �� ( )

( ) ( )

3

3

2013( 2010). 2011 2012. 1

2010 . 4024 2012

y

x

x

y

− + =

− =

=

�. �ក�គប"ប,- ច�ន�នប[ម , ,e n h �./ម0� ! enh en nh he< + + �

1. �គ !�ងBង"ផa�� I � R ន�ងច�ន�ចនDង A �2�5/�ងBង"� ព�ន��!ប,- )ខpធ�* BC �ប"�ងBង"

�ផ7GងH7 �"5កq:r 2 2 2A AC kB BC+ − = , ).5 k 'ច�ន�ន).5�គ !� �ក�ន��ច�ន�ច

ប,- ច�ន�ចក,- 5 M �ប" BC (ព�iកy�*ប^ង�ន��ច�ន�ច=ម ,k R )�

Page 114: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 113

E. �គ !ប�ច�ន�ន# �ជ%&ន , ,a b c �

��យJ� a b c a b c

a b b c c a b c c a a b+ + < + +

+ + + + + + �

K. �ក�ម�យទ�ព<�&ន�� មទDកជក"ម�យ).5&ន�ក�Cផ7ធ�'ង 1� ��យបI% ក"J,

�គbច)ចក�ក�ទ�ព<�'ប,- ���ក= 89ង, ! ម�ន&នក�ព*5,�ប" ��

o3 ក"ច*5�Pក��ងក)ន3ង&នទDកជក"�

N. �គ !�F5�ប� 2 2

1 22 21,: .( )

x yE

a bF F+ = 'ប,- ក�ន��� M ��"�2�5/�F5�ប ( )E �

ប67 �"ព���ប"ម�� �1 2F MF �" 1 2F F ��ង" ,N H 'ច��,5)កង�ប" N �2�5/ 1MF �

��យបI% ក"J MH ម�ន)�ប�ប|5 �

'()&'()&'()&'()&

ចំេល�យ

�. �បព<នQម� �).5 !មម*5នDង�បព<នQ�ង�� ម�

( )( )( )

( ) ( )( )3

3 3

33 3

2010 2013( )

201

20

0 201

11 2012 1

20123 2011

x yI

x y

− −

− − −

+ = =

ព�ន��! 2010x = ម�ន)មន'U�ប"�បព<នQ�6�

( )( )

33

3

33

3

20132010

2013 2011

12011 2012

( )

2012.201

1

0

yx

yx

I

− =−

− − =

+

=ង� 3

3

2013

1

2010

y

vx

u −

=

=

� �បព<នQ 3 យ�P'�

( )( )2 23

3 3

20122012. 2011

2012. 2011 201

0

2. 2011

u v uv

u

uv vu

u v v

− + + += +⇔

= + = +

=

Page 115: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 114

(An នU)

3

2 2

3

2012 2011 0

2012 0

2012. 2011

u

u

uv v

u v

v

u

u

− − =

=

⇔+

+ + =

= +

1

1 8045

2

u v

u

u

= = − =

⇔±

3

3

1

1 8045

2

2013 1

11

2010x

u

u v

yv −= = − = =

= −⇔ ⇔± = −

� M 3

3

2013

201

1 8045

2

1 1 804

20

5

y

x

+

+ =

− =

� M 3

3

2013

201

1 8045

2

1 1 804

20

5

y

x

− =

− =

2009

2012

x

y

= =

� M 3

3

22010

1 8045

1 80452013

2

x

y

= +

+ += +

� M 3

3

22010

1 8045

1 80452013

2

x

y

= +

− −= +

�. ��យប,- ច�ន�ន ,,e n h &ន�ទQ�n/A� �6���យម�ន�ធB/ !;�"បង"5កq:�ទ*�P,

�យ/ងzប&J e n h≤ ≤ , Sញ;ន 3en nh h he n+ ≤+

�ប/ 3 3 ,e en nhnh enh enhhe≥ ⇒ ≤ ≤+ +⇒ ផ7�យព�ប�^ប"�បoន

.*ច�ន� 2e = (��Z� e 'ច�ន�នប[ម)�

6� !� 1 1 12 2 2 5

2nh n nh h n

h n⇒+ + > ⇒< + <

+ 2n h= ⇒ 'ច�ន�នប[ម,កL;ន

3n h+ = ⇒ 3= � M 5h = �

.*ច�ន� ច��5/យ�ប"5�l�"គM� 2,2,n h pe = == 'ច�ន�នប[ម ន�ង�គប"ច��"�ប"?,

� M 3 32, ,n he = == ន�ង�គប"ច��"�ប"?�

1. =ង J 'ច�ន�ចក,- 5 AI R= (.*ច�*ប)

�យ/ង;ន� 2

2 2 2 (1)22

BCA CB AMA =+ −

2

2 2 22.2

(2)IA MR

M JM+ − = B

A

C

D

E

F

J I

M

Page 116: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 115

=មប�^ប" 2 2 2 (3)AC BCAB k+ − =

=ម (1)ន�ង (3)�យ/ង;ន� 2

2 2 2 2 (4)22 2

BC kAM k AM IM R− ⇔ + − ==

ព� (4)ន�ង (2) �យ/ង;ន� ( )2 21( )

45JM R k= +

ច�,�� (3) (5)⇔ �6��គប"ច�ន�ច M ).5��e#�ក ��e#4���2�ងក��ង�ងBង" ( , )I R ន�ង

�ផ7GងH7 �" (5)�

.*ច�ន�

∗ �ប/ 2k R< − �6��ន��ច�ន�ច).5��e#�ក '�ន��ទ�ទ�

∗ �ប/ 2k R= − �6��ន��ច�ន�ច).5��e#�ក 'ច�ន�ច J �

∗ �ប/ 2k R> − �6� M 4���2ក��ង�ងBង" ( , )I R �O/យ M 4���2�5/�ងBង" ( , )J r ,

ច��Z� 21

2r kR= + �

5�ម���ប"�ន��ច�ន�ច�

� �ប/ 28k R≥ , �ព5�6� 3

2r R≥ , �6��ន��ច�ន�ច).5��e#�ក'ច�ន��ទ�ទ�

� �ប/ 20 8k R< < , �ព5�6� 3

2 2

Rr R< < , �6��ន��ច�ន�ច).5��e#�កគM'ធ�*�ងBង" �EDF �ប"

�ងBង" ( , )J r ម�នគ��ព��ច�ន�ច ,E F ( ,E F 'ច�ន�ច�បពB�ប"�ងBង" ( , )J r ន�ង�ងBង" ( , )I R �O/យ

D 'ច�ន�ច�បពB�ប"�ងBង" ( , )J r ន�ងកន3�ប67 �" AI )�

� �ប/ 0k = , �ព5�6� 2

Rr = , �6��ន��ច�ន�ច).5��e#�កគM'�ងBង" ( , )J r ម�នគ��ច�ន�ច A �

� �ប/ 2 0kR− < < , �ព5�6� 2

Rr < , �6��ន��ច�ន�ច).5��e#�កគM'�ងBង" ( , )J r �

E. .�ប*ង, �យ/ងfយនDង��យ;នJ� ច��Z� , ,x y m 'ប,- ច�ន�ន# �ជ%&ន89ង, !

1x

y< �6� x x m

y y m

+<+

Fន�#�-នj5ទQផ5�ន� �យ/ង;ន�

, ,a a c b b a c c b

a b a b c b c b c a c a c a b

+ + +< < <+ + + + + + + + +

ប*កFងRនDងFងR Cន# �មiពS�ងប��ង�5/ �យ/ង;ន�

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 116

(12 )a b c

a b b c c a+ + <

+ + +

មu9ង�ទ@�� ,,a b c # �ជ%&ន�6� �យ/ង;ន� 1( ) ( )

2a b c a b c+ + +≤

Sញ;ន� 1 2

( )

a aa

b c a b c a b c=

+ + + +≥

.*ចA� ).�� 1 2

( )

b bb

c a b c a a b c=

+ + + +≥ , ន�ង 1 2

( )

c cc

a b c a b a b c=

+ + + +≥

ប*កFងRនDងFងRCន# �មiពS�ងប��ង�5/ �យ/ង;ន�

2 (2)a b c

b c c a a b+ + ≥

+ + +

ព� (1) ន�ង (2) �យ/ង;នបIg ��e#��យបI% ក"�

K. �យ/ងង"=^ង�,ញ"^ង' ���ក=,ម�យ ��ច �"H- ច"ប,- �ប��� ���ចញ

ព��ក�� �យ/ង��@បប,- �ប��� ���ន� ��|�គ�ជ���5/A� � zប&J �� មទDក��n

bច�'បឆ3ង �"ន3Dក�ក�� .*ច�ន� �2ន3Dក�ក��2�ង�5/បង�" នDង��e#

ប,- �� មទDក��n �'ប�k/ង 3 យ�P'�� មទDក��n ម�យ ).5&ន�ក�Cផ7�*ច'ង 1

(ប,- �� មទDក��n bច�2�'ប�2�5/A� )� 6� !&ន ច�ន�ច P �ប" ���2�5/បង�"

ម�ន&ន�� មទDក��n ,S�ងF"� �យ/ងយកម%�5ម�យ �\�ទ�5���គប"ន3Dក�ក�).5

��|��5/A� �6� ព��5/ច���� យ �"=មច�ន�ច P � យកប,- �*ប �� �P^យ����@ប=ម

ទ�=�ង\"# �ញ �6�ប,- �ប����ប�lង�ន� ប�ង̀/�;ន'=^ង�,ញ" ��ម�យ ).5ម�ន&នក�ព*5, 4���2ក��ង�� មទDក��n S�ងF"�

N. .�ប*ង �យ/ង��យបI% ក"5ទQផ5).5�យ/ងo3 ប".Dង'ម�ន�

ក��ង���� : ABC ( ,,a b c '�ជmងS�ងប�), ច��Z�ប67 �"ព�� ,AD I 'ផa���ងBង"\� Dកក��ង,

�យ/ង;ន� AD a b c

AI b c

+ +=+

ព��'.*ច�ន�, �យ/ង.DងJ� 0 (1)IA bIB ca IC+ + =� � � �

=មចtប"ច��,5#� �ចទ<� =មទ� BC �P�5/ប67 �" 1), (AD 3 យ�P'�

0 ( ) 0a aIA bIB cID IA b c ID+ + = ⇒ + + =� � � � �� �

.

Page 118: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 117

Sញ;ន� ID a AD a b c

IA b c AI b c

+ += =⇒+ +

��kប"មក5�l�"# �ញ, �យ/ង&ន 1 2 1 2

1 2

MFMFMN

MI

F F

FMF M

+ ++

= (=ម5ទQផ5�ង�5/),

1 2 1 2

2

MF FFMK

FM + −=

ព��6� 1 2 1 2 1 2 1 2

1 2

.2

MF FMF MFMH MNMH MK MK

MK MI MF

F MF F F

MF

+ + + −+

= = =

2 2 22 2 2 2 ( )( )

.2 2

a c a c a c a c a b

a

c

a a a

+ − −+ −= = = = ម�ន)�ប�ប|5�

'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២៣

�. �����យ�បព<នQម� �� 3

(1)

2 1 (

1

2

1

)y

xx y

x

y − =

+

=

�. �កU'ច�ន�នគ�"�ប"ម� �� 2( 1)( 7)( 8)x x x x y+ + + =

1. �គ ! ABC∆ &ន�ប)#ងប,- �ម.uន ន�ង ��ងBង"\� Dក��]��@ងA� គM , , ,a b cm m m ន�ង

R � ��យបI% ក"J� 9

2a b cm mR

m + + ≤ �

E. ��យបI% ក"J ច��Z��គប"ច�ន�ន# �ជ%&ន , , ,a b c d �ផ7GងH7 �"5កq:r 4a b c d+ + + =

�គ;ន� 2 2 2 21 1 1 1

2a b c d

b c d bac d a+ + +

+ + +≥

+

K. �គ !�ន�� .{1;2;3;4; .. ;99;100}X = �O/យ A'�ន���ង&ន 51o���ប" X �

B

A

C

I

D

y

O

H K

M

1F 2F x

I

N

Page 119: Vnmo 30 4-2011-grade 10

�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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��យបI% ក"J &នព��o���ប" A'ព��ច�ន�នប[ម�?ងA� �

N. ��យបI% ក"J ម�នbច&ន�2�5/ប3ង"ក*F����ន ន*#ច��� : ABCD ).5

( ) 0, 6,2 3. 0AAC BD C BD= =� �

�O/យក*F����នប,- ក�ព*5�ទQ)�'ច�ន�នគ�"�ទ�

'()&'()&'()&'()&

ចំេល�យ �. 5កqខ:r � 0xy ≠

ច��Z�5កqខ:r �ង�5/, ( ) 1(1) 1 0

1

x yx y

xyxy

= − + = = −

�����យ�បព<នQ� 3 3

1

1 52

2

1 5

2

1 2 1 0

x y

xx y x y

xy x xx

x

= == =

− + ==

− −

⇔ ⇔+

=

− + =

�����យ�បព<នQ� 3

3 4

1 11

21 0

2 12

yxy yx xy x

x x xx

⇔ ⇔+ +

= −= − = −

= − = +

+ =

2 22

1

1 1 30

2 2 2

yx

x x

= − + =

+

− +

�បព<នQAn នច��5/យ

��បGប�ធ@បនDង5កqខ:r 0xy ≠ �6��បព<នQ).5 !&នច��5/យប�គM�

1 5 1 5

1 2 2; ;1 1 5 1 5

2 2

x xx

yy y

− + − −= = = = − + − − = =

�. ព�ន��!ម� �� 2( 1)( 7)( 8)x x x x y+ + + = (1)

�យ/ង&ន� 2 2 28 )((1) 8 )( 7x x x x y⇔ + + + =

2 2 2 28 ) 7(( 8 )x x x x y⇔ + + + =

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2 2 2 28 ) 284( ( 8 ) 4x yx x x⇔ + + + =

( ) 22 28 4 492 7x x y + −⇔ + =

( )( )2 216 2 7 16 2 7 (22 49 )2x xx y x y⇔ + + + + =− +

ព� (2)Sញ;ន 6 �បព<នQម� ��ង�� ម (ច�,� ,x y ∈ℤ )

2

2

16 2 7 49 (3)

2 16 2 7 1)

(

2

4)

x y

x x y

xa

+ + + =+ − + =

ព� (3)ន�ង (4)Sញ;ន 4 48y = � M 12y = , ជ�ន�ច*5 (3)�យ/ង;ន�

2 216 18 0 81

2 99

0x xx

x xx

+ − = ⇔ + − === −

.*ច�ន� ក�:� )a &នច��5/យព�� (1;12) ន�ង ( )9;12−

2

2

16 2 7 1

2 16 2 7

2)

49

x yb

y

x

x x

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2

2

16 2 7 492 4)

122 16 2 7 1

x y

x x y

x xc

y

+ + + = −⇔

+ − + = −

= − = −

2

2

16 2 7 1

2 16 2

4

1249

2)

7

x y

x x

x xd

yy

+ + + = −⇔

+ − + = −

= − =

2

2

16 2 7 7

2 16 2

0

02

0

7 7)

8

x y

x x y

x

yxe

x

y

= =

+ + + =

= −

=

+=

+ −

2

2

16 2 7 7

2 16 2 7 7

1

02)

7

0

x y

xf

x

x

x

y

y

x

y

= − = + + + = − ⇔

+ − + = −

= − =

.*ច�ន� �បព<នQម� �).5 !&នប,- ច��5/យ'ច�ន�នគ�"គM�

12), ( 9;12), (1; 12), ( 9; 12), ( 4; 12), 12), (0; 0), ( 8; 0),(1; ( 1; 0), ( 7; 0( ; )4− − − − − − − − −− �

1. =ង O 'ផa���ងBង"\� Dក��] ABC∆

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�យ/ង;ន� ( )20OA OB OC+ + ≥

� � �

( )2 2 2 2 . . . 0OB OC OA OB OB O OO C C OA A⇔ + + + + + ≥� � � � � �

2 22 (cos2 cos2 cos2 )3 0R A BR C⇔ + + + ≥

2 2 2 2 22 (3 2sin 2sin 2sin ) 03R R A B C⇔ + − − − ≥

2 2 2sin9

sin4

sin (*)A B C⇔ + + ≤

មu9ង�ទ@�, =ម# �មiព Bunyakovski , �យ/ង;ន�

2 2 23 )(a b c a b cm mm m m m+ + ≤ + +

2 2 23. (4

)3 a b c+ +=

2 2 2 2(sin s s9 in in )A B CR + += 2 9 49

4 2.R R≤ = (=ម (*) )

.*ច�ន� 9

2a b cm mR

m + + ≤ , Iw " "= �ក/�&ន5���=)����� : ABC ម<ងp�

E. =ម# �មiព Cauchy �យ/ង;ន�

2 2

2 21 1 2

ca ab aba

b b bc c

ca

c= −

+ +≥ −

)� 2 . . ( )

2 2 42

ab ab c b a a c b a aca a a

b c

ca

+− = − = − ≥ −

6� ! 2

1( )

1 4

aab a

b cbca +≥ −

+

.*ចA� ).� �យ/ង��យបI% ក";ន�

2 2 2

1 1 1( ); ( ); ( )

1 4 1 4 1 4

b c dbc bcd cb c d

d ad cda da dab

c d ba+ + +

+ + +≥ − ≥ − ≥ −

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2 2 2 21 1 1 1

a b c d

b c d bd ac a+ + +

+ + + +(1)

1( )

4ab bc cd abc bcd cda daba b c d + + + + + +≥ + + + −

មu9ង�ទ@�� 21( )

44 (2)ab bc ca da a b c d+ + + + =+ +≤

2 216( ) 16 4( ) ( ) 4( )( 6 ( ) () 1 )abc bcd cda dab ab c d c a b c d c d aa bd b+ + + = + + ≤ + + + ++ +

34( )( )( ) ( )a b c d a b c d a b c d≤ + + + + + ≤ + + +

B

A

C

a b

c

O

Page 122: Vnmo 30 4-2011-grade 10

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� M 31( )

14 ( )

63abc bcd cda dab a b c d+ + + =+≤+ +

.*ច�6�, ព� (2),(1), (3), �យ/ង;ន�

2 2 2 2

1(4 4) 4 2 2

1 1 1 1 4

a b c d

b c d aa b c d

c d a b+ + + + = − =

+ + + +≥ + + + −

មiព�ក/�&ន5���=)� 1a b c d= = = = �

K. )ចក�គប"o���ប" X ' 50គ* (3;4), ..., (99( ;1;2), 100)� ��យ A &នផ7�ក 51o���ប" X

�6�&ន89 ង��ចម�យគ* ( ; 1)k k + ក��ងប,- គ*�ង�5/ ).5��e#;ន��ជ/យក, )� k ន�ង

1k + 'ព��ច�ន�នប[ម�?ងA� �6��យ/ង;នបIg ��e#��យបI% ក"�

N. �យ/ង��យបI% ក"��យ# �ធ�ផ7�យព� �ព��

zប&J &នច��� : ABCD &នក*F����ន�គប"ក�ព*5 �ទQ)�'ច�ន�នគ�" �O/យ

( ) 0, 6,2 3. 0AAC BD C BD= =� �

�ព5�6�, �យ/ង;ន�

. ( )( ) ( )( ) (1)C A D B C A D BAC BD x x x x y y y y= − − + − − ∈� �

ℤ .

មu9ង�ទ@�� 0 2. . .cos( , ) 2 3 . .cos60 3 (2)AC BD AC BD AC BD BD BD BD= = = ∈/� � � �

(��Z� 2 2 2( ) ( )D B D BBD x x y y= − + − ∈ℤ )

�យ/ង�ឃ/ញJ (1)ន�ង (2)ផ7�យA� �6� �zប&គMខ��

.*ច�ន� 5�l�"��e#;ន��យបI% ក"��ច^5"�

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វ��� �ទី២៤

�. �����យម� �� 2 4 2 413 9 16x xx x ++ =−

�. ច*�បក��យJ ��/&នច�ន�នច�ន�នប[មម�យ� M�ទ�2ក��ងB����ង�� ម� 41 ,x+

4 8 4 4, ... ,1 ..1 . nx x xx + + ++ + ច��Z� ,x n 'ច�ន�នគ�"ធ�'ង 1, x 'ច�ន�ន��

1. �គ ! M 'ច�ន�ច,ម�យ4���2ក��ងប3ង"���� : ABC �

�ក��C5�*ចប�ផ���ប"� MA MB MCT

a b c= + + �

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E. �គ ! 1 2, , ..., (0; ]naa a p∈ ច��Z� 0p > � ��យបI% ក"J� 1

1 21 2

( )( ) ... ( )...

n n

nn

p pp a p a p a

aa a n

+≥

+ ++ − −−

+ ច��Z� *n∈ℕ �

K. )ខp�ច?9ក"ម�យ&ន 2011J� �ង, J� �ង�ច?9ក"ន�ម�យ�&នទ�ងន" 1� ម� ��/��e#���ប9�6n ន

J� �ង�ច?9ក" �./ម0� ! �ព5យកប,- )ខp�ច?9ក"�6��ធB/'ជ�% �ង គMbចថ3Dង (�2�5/J

ជ�% �ង) ប,- #�4�).5&នទ�ងន"\ប"ព� 1�P.5" 2011� ម �គ.Dងច�ន�នJ� �ង�ច?9ក"��e#

����ចញ89 ង��ចប�ផ��?

N. ក��ងប3ង" �គ !ប67 �" ∆ ន�ងច�ន�ច A∈ ∆/ � ព�ន��! ,B C ∈∆ 89ង, !�ប)#ងFង̀�" BC

ម�ន)�ប�ប|5� �ក�ន��ច�ន�ច ផa���ងBង"\� Dក��]���� : ABC � '()&'()&'()&'()&

ចំេល�យ �. 5កqខ:r 11 x− ≤ ≤

�5/កFងRS�ងព���ប"ម� �' �� �យ/ង;ន�

( )22 2 213 1 9 1 256 (1)x x x− + + =

Fន�#�-នj# �មiព Bunyakovski , �យ/ង;ន�

( ) ( )( ) ( ) ( ) ( )2

2 2 2 2 213. 13 1 3 3 3 1 13 27 13 13 40 13 03 16x x x x x≤ + +− + + + − = −

FងR�ង�ឆBង�ប"ម� � (1) ( ) ( )2

2 2 2 2 2: 13 1 9 1 16 1040x x x x x≤− + + −

Fន�#�-នj# �មiព :Cauchy ( ) ( )2

2 2 2 264 2516

10 16 1 60 40 6 102

x x x x − − = ⇔ ≤

Iw " "= �ក/�&ន�ព5 ( ) ( )2 2

2 2

13 1 3 12

13 3 3 51 10 01 6

x x

x

xx

=

− + = = −

� M 2

5x = −

�ន��U�ប"ម� �គM� 2 2,

5 5S

=

�. ច��Z� n �� =ង *(1 )2 mn m + ∈= ℕ

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�យ/ង;ន� 4 4 4 8 4... 1 ..1 .n mxx x x ++ + = + + ++

( ) ( ) ( )4 8 4 8 41 1 . 1. . mx x x x x= + + + + ++

( )( )4 8 8..1 1 . mx x x+ += + + 'ច�ន�ន&

ច��Z� n គ*� =ង *( )2 mn m ∈= ℕ

�យ/ង;ន� 4 4 4 8...1 1 ...n mx x x x+ + = + + ++( )( )4 8 8

4

.1 1 ..

1

mx x

x

x+ +− +=

( )( )( )( )

4 2 4 28 4 4 2 4 2

4 2 22 2

1 11 1 1.

1 1 11 1

m mm m mx xx x x

x x xx x

+ ++ + +− +− − += = =− − +− +

( )4 2

2 42

.1

..11

mm x

xx

x+ += +

++

+

'ច�ន�ន&�

ន���h ន� ម�ន&នច�ន�នប[មក��ងB����ង�5/�ទ�

1. =ង G 'ទ��បជ��ទ�ងន"�ប"���� : ABC �6��យ/ង;ន�

. . .

. . .

MAGA MB GB MC GCT

a GA b GB c GC= + + 3 . . .

2 . . .a b c

MAGA MB GB MC GC

a m b m c m

= + +

=ម# �មiព Cauchy �យ/ង;ន�

2 2 2 2 2 2 21 12 )(2. 2 (3

2 22 )

3aam a b ac a b c a+== + − −

2 2 2 2

2 2 2(2 2 ))

1 3 1. .(

22 3 2 3

b c ab c

aa

+ +≤ +=− +

មiព�ក/�&ន5���=)� 2 2 2 2 2 2 22 23 2b c a ca b a= + − ⇔ + =

��យបI% ក".*ចA� ).� ច��Z� ,b cbm cm �

.*ច�ន� ( )2 2 2

3 3. . .T MAGA MB GB MC GC

ba c+ +≥

+ +

មiព�ក/�&ន5���=)� 2 2 2

2 2 2

2 2 2

2

2

2

b c

c ab

ac b

a + = + =

+ =

⇔ ���� : ABC ម<ងp�

មu9ង�ទ@�� . . . .. .MAGA MB GB MC G MAGA MBC GB MC GC≥ + ++ +� � � � � �

( ) ( ) ( ). . .MG GA GA MG GB GB MG GC GC+ + + += +� � � � � � � � �

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( ) ( )2 2 2 2 2 2 2 2 24 1

9 3a b cGA mGB G aC m m b c+ + = + +=+ +=

មiព�ក/�&ន5���=)� �គប"#� �ចទ<� , ,MA MB MC� � �

��@ងA� &នទ�.*ចនDង ,GA GB� �

,GC�

.

&នន<យJ ច�ន�ច M ��|�A� នDង G �

.*ច�ន� min 3T = �ព5���� : ABC ម<ងp �O/យ M ��|�A� នDង G �

E. zប&J ( )1 1 2max , , ..., na aa a=

2 3 2 3) ( ) ... ( ) ( ... )(1 n np a p ap a p a a a

np

+ − + + − + + + + +−=

2 2 3) ... ( )( ... )( n np a p aa ap a

p

− + + + +≥

2 2 3( ) ... ( )( ... )nn np a p p a ap a a⇔ ≥ − − + + + +

1 1 2 2 3( ) ( )( ) ... ( )( ... )nn np a p a p a p a p a ap a⇔ − ≥ − − − + + + +

11 2

2 3

( ))( ) ... ( )

...(

n

nn

p ap a p a

pp a

p a a a

−⇔ − − ≤+ + +

−+

1

1 2 3

( )

...

n

n

p a

a

p

a a a

−≤+ + + +

(��Z� a p≤ )

1 1

1 11 2

1 2 3 1 2 3 1

)( ) ... ( )... ...

(n n nn

nn n

p p a p app a p a

a a a a a a a ap

naa

+ +−⇔ − − ≤ ≤ −+ + + + + + + +

1

1 21 2 3

)( ) ... ( )..

(.

n n

nn

pp a p a

a a

pp

n a aa

+

⇔ − − ++ +

− ≤+ +

(បIg ��e#��យបI% ក")

K. zប&J ច�ន�នJ� ��ច?9ក" ).5��e#�����ចប�ផ�� �./ម0��ផ7GងH7 �"5�l�"គM k �6��យ/ង

ថ3Dង;នព� 1�P.5" k � ម (ច�,�� J� �ង�ច?9ក"ន�ម�យ�).5����ចញ &នទ�ងន" 1� ម

�n/A� ��ប/�./ម0��ធB/')ផ3ជ�% �ង)�

.*ច�6� J� �ង.�ប*ង).5�យ/ង����ចញ គM'J� �ងទ� 2k + �./ម0� !;នFង̀�"&ន�ប)#ង

1k + J� �ង� �ព5�6� �យ/ងថ3Dង;នព� 1�P.5" 2 1k + �

បន-�ទ@� �យ/ង��e#���J� �ង�ច?9ក"ទ� 3 4k + �./ម0�;នFង̀�"&ន�ប)#ង 2 2k + J� �ង

(��Z� (2 1) 1 ( 2) 3 4k k k+ + + + = + , �5ខ 1�./ម0�បfg ញJ� �ង).5នDង��e#����ចញ)�

បក��យ.*ចA� ).�, �យ/ង�ឃ/ញJ Fង̀�"J� �ង�ច?9ក" នDង��e#;ន)បង)ចក.*ច�ង�� ម�

1, 2( 1), ..., 2, ( 1)kkk k k+ + + �

.*ច�ន� k 'ច�ន�ន�*ចប�ផ���ផ7GងH7 �"� 1. 2011 2 ( ) 12 1k kk k+≤ ≤ + − គM 7k = �

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�ព5�6�, �យ/ង;ន 7Fង̀�"J� �ង�ច?9ក"�ប)#ង 16, 32, 64,128, 258, 6, 512J� �ង

(��e#A� នDង 7J� �ង).5����ចញ, ម�យJ� �ង�&នទ�ងន" 1� ម)�

N. ��យម�ន�ធB/ !;�"5កq:�ទ*�P, \�"ទ�កJ 2BC =

=ង O 'ច��,5)កង�ប" A �P�5/ ∆ �O/យ=ង ( );a d A= ∆ �

��ជ/�� /����យក*F����ន Oxy 89ង, ! 0), (( ; 0; 0)A Oa (គMJF<កpbប"��&នផ7�ក

∆ , F<កpF����ន&នផ7�ក OA )

zប&J ក��ង�បព<នQ����យ�ន� 0 0; 0), ( 2; 0)( Cx xB + (��Z��ប)#ង 2BC = )�

=ង H 'ច�ន�ចក,- 5 BC , �ព5�6� 0 )( 1; 0H x + ន�ង 1HB HC= =

=ង ( ; )I x y 'ផa���ងBង"\� Dក��]���� : ABC , �ព5�6� 0 1x x= + ន�ង IA IB=

Sញ;ន 20 1 1( )2ay x + −= � M

2

2*

1

2( )

xy

a a= −

.*ច�ន� �ន��ច�ន�ចផa���ងBង"\� Dក��]���� : ABC ';9 9̂ប*5&នម� � (*) �

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វ��� �ទី២៥

�. �����យម� �� 4 4 2 22012 1011

20112010

x x xx + =+ +

�. �គ ! 4ច�ន�នគ�"�A� � ^5"�ព5Fន�#�-នjម�យ�5/ក�, 5�ប�\5ព��ច�ន�ន ,a b ,

កL��យ ��ចជ�ន���យ a ab+ ន�ង 3b ab+ � ��J��/ bច&ន�ព5,�6�).5

�គ;ន 4ច�ន�នគ*�A� � M 4ច�ន�ន��A� � M�ទ?

x

A I

H B C O y

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1. �គ !���� : ABC �2នDង �O/យប67 �" d ច5<� �"=ម C � ��e#A� នDងទ�=�ងន�

ម�យ��ប" d , �គ�dច�ន�ច M �2�5/ d 89ង, ! 2 23 5MA MB+ &ន��C5�*ច

ប�ផ��� �ក�ន��ច�ន�ច ច�ន�ច M �

E. �គ !ប,- ច�ន�នព��# �ជ%&ន , ,a b c � ��យបI% ក"J�

2 2 2 2

9( )

( )ab bc bc ca c

a b c ab bc ca

a b c ca a bab

+ ++ +++ + + + +

≥+ +

K. �ន�� X &ន 2n o����e#;ន)ចក�ចញ'ប,- �ន���ង).5ម�ន�បពBA� ព�ម�យ�P

ម�យ� ព�ន��!# �oនក��ង �H3 "ប-��o��A� .*ច�ង�� ម� �ប/ ,A B 'ប,- �ន���ង�ប"

X ,�O/យច�ន�នo���ប" Aម�ន��ច'ងច�ន�នo���ប" B �6��គ&ន�ទQក��ង �H3 "

ប-*�ព��ន�� Aច*5�ន�� B ន*#ច�ន�នo���n/នDងច�ន�នo���ប" B � ��យបI% ក"J

�� យព�ជ�lនH3 "ប-��)បប�ន�ប9�6n ន.ង�P �គទទ�5;ន�ន�� X �

N. �គ !�F5�ប� 2 2

2 2: () 1 )( 0

x yE

a ba b> >+ = � =ង 1 2,A A 'ក�ព*5S�ងព���2�5/F<កpធ�

ន�ង M 'ច�ន�ចច5<��2�5/ ( )E � =ង H 'F��*ង"���� : 1 2MA A �O/យ K '

ច�ន�ចឆ3��នDង H �"=មប67 �"ព���ប"ម��ម�យiគប�នទ�ម�យ, ទ�ប��

�ក�ន��ច�ន�ច ប,- ច�ន�ច K � '()&'()&'()&'()&

ចំេល�យ

�. ជ�ន� 2011 a= , �យ/ង;ន�

4 4 2 2 ( 1)x aax x x a a++ + = −

4 4 2 2 2 0x aax x x a a⇔ + + + − =+

( ) ( )( )4 2 2 2 0ax a x x a x aa a⇔ + + ++ + − + =

( )( )2 4 2 4 20 0a x x x aa a x aax⇔ + + + ⇔ + ++ − = − =

4 2 2 21 10

4 4x axxx a⇔ + + − −+ +− =

Page 128: Vnmo 30 4-2011-grade 10

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Page 127

( )( )2 2

2 2 2 2 2 210 01

1

2 2ax x x x x axa ⇔ + − + = ⇔ + + +− =

− +

2 2 4 21 10 0ax x x x a⇔ + − + ⇔ + + − ==

2 1 4 30

2

ax⇔ = − + − >

.*ច�ន� ម� �).5 !&នUព�� 1 8041

2x

− +±=

�. ផ5ប*កCន 4ច�ន�នគ�"�A� � 1 2 3 4 6 (mod2 4)x x x x x+ + + + + + = ≡+

��យ 3 (mod 4)a ab b a a bb ≡ ++ + + �6��� យ�ព5 Fន�#�-នj �ប*ក�គប"ច�ន�នម-ង�

គM�2)�)ចកម�ន�ច"នDង 4�

មu9ង�ទ@�, បក��យ.*ចA� នDង�ង�5/).�, ផ5ប*ក 4ច�ន�នគ*�A� � M 4ច�ន�ន��A�

)0 (mod4≡ �

.*ច�6�, ម�នbច&ន 4ច�ន�នគ*�A� � M 4ច�ន�ន��A� �ផ7GងH7 �"��:/ ��បoន�ទ�

1. �2�5/�ជmង AB �dច�ន�ច D �ផ7GងH7 �" 3 5DA DB= �

=ម�ទD-�បទក*��ន�, �យ/ង;ន�

�2 2 2 2 . .cosDA DM DA DM DMMA A= + −

�2 2 2 2 . .cosDB DM DB DM DMMB B= + −

2 2 2 2 25 3 5 83 MB DA DM A DB M⇒ + = + +

( )2 2 23 min 85 minMA DB MM dDM⇒ + ⇔ ⇔ ⊥

⇒ �ន��ច�ន�ច M '�ងBង"Fង̀�"ផa�� DC �

E. # �មiព).5 !មម*5នDង�

2 2 2

2 2 2ab bc b

a b c

c cb abca caa + + ++ +

+ + ++

2 2 2 2

( ) ( ) ( ) 9( )

( )

a b c b c a c a b ab bc ca

a b c a bab bc bc ca ca a cb≥

+ + + ++ + + + ++ + +

++ ++

�យ/ង&ន� 2 2 2 2

2 2 2 2 2 2

)

21

( )

(

ab bc bc ca ca ab

a b c a b c

a b b c ab ba cac c≥

+ + + + + + + + + + ++ ++ + =

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Page 128

2 2 2

( ) ( ) ( )

ab bc bc ca ca

a b c b c a c a b

a c abb+ + +++

+ ++

++ +

2 2 2 2 2 2

2 2 2

( ) ( ) ( )

)( )( ( ( ( () ) ))

a b c

a b c a b c a

b c c a a b

ab bc bb c a b cc ca ca ab= + +

+ ++ +

+ + +++

+ + +

( )( )( )

( )2

2 2 22 2 2

4 4

b c ab bc cab c ab b

ab bc ca ab bc ca

aab bc ca c caa

+ + + +=

+≥

+ + + ++ + ++ ++ +

�យ/ង��e# ���យបI% ក"� ( ) ( )( )22 2 2

4 91

b c

ab

ab bc

bc ca ab bc ca

a aa cc b≥

+ + + ++ +

++ +

++ +

2 2 2 2

1 4 9

( )b c ab bc cab bc ca a a b ca+

+ + ++ + + + +⇔ ≥

+

2 2

2 2 2 2

1 2 9

( )b c ab bc cab bc ca a a b ca⇔ ≥

+ + ++ + + +++

+ (ព��)

.*ច�ន� # �មiព).5��e#��យបI% ក"គMព��, Iw " "= �ក/�&ន5���=)� a b c= = �

K. ច�ន�ន�ន���ង &នច�ន�នo��'ច�ន�ន� ��e#)�'ច�ន�នគ* ��Z� 2n 'ច�ន�នគ*, zប&J

&ន 2k �ន���ង).5&នច�ន�នo��'ច�ន�ន�, )ចកព�ក?' k គ* ��ចFន�#�-នjចtប"

ប��"ទ�o�� �យ/ងនDង;ន�P.5"ក�:� ).5 �គប"�ន���ងS�ងF"�ទQ)�&នច�ន�នo��

'ច�ន�នគ*�

�ព5�ន�, ច�ន�ន�ន���ង).5&នច�ន�នo��'ច�ន�នគ* )�)ចកម�ន�ច"នDង 4 ��e#)�'ច�ន�នគ*

��Z� 2n )ចក�ច"នDង 4 , zប&J &ន 2m �ន���ង).5&នច�ន�នo��'ច�ន�នគ* )�)ចក

ម�ន�ច"នDង 4 , )ចកព�ក?' m គ*��ចFន�#�-នjចtប"ប��"ទ�o�� �យ/ងនDង;ន�P.5"

ក�:� ).5�គប"�ន���ងS�ងF"�ទQ)�&នច�ន�នo��)ចក�ច"នDង 4�

Fន�#�-នj.*ចA� , �� យមកប9�6n ន.ង, �យ/ងនDង;ន�P.5"ក�:� ).5�គប"�ន���ងS�ងF"

�ទQ)�&នច�ន�នo��)ចក�ច"នDង 2n �O/យ�ព5�6� �យ/ងទទ�5;ន�ន�� X �

N. =ង ( , ) ( )M x y E∈ ន�ង P 'ច��,5)កង�ប" M �P�5/ 1 2A A

�យ/ង;ន� 1A PH∆ ន�ង 2MPA∆ &ន^ង.*ចA� , 6� !�

2 2 2 211 2

2

APHPH

PPM PA A

PA MPP= ⇔ =

( ) ( )2 22 2Hy a ay x x⇔ += − ( ) ( )

222 2 2 2 2

2H

by a x a

ax⇔ − −=

Page 130: Vnmo 30 4-2011-grade 10

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( )2 2 2 2 2H Hy a a xb⇔ = − (��Z� Hx x= )

2 2

42

2

1H Hx y

aab

⇔ + =

��យ K ឆ3��នDង H �ធ@បនDងប67 �"ព���ប"ម��ម�យiគប�កទ�ម�យ, ន�ងទ�ប�

6� ! ;H K H Kx yy x= = Sញ;ន 2 2

4 2

2

1K Kx y

a ab

+ = �

.*ច�ន� �ន��ច�ន�ច K '�F5�ប&នF<កpធ��2�5/ Ox �ប)#ង2

2a

b, F<កp�*ច�2�5/ Oy &ន

�ប)#ង 2a �

'()& វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២៦

�. �កប,- ច�ន�នគ�"ធមn'�� ,a b �./ម0� !ច�ន�ន 29 3 863 10a bA + −= + 'ច�ន�នប[ម�

�. �ក m �./ម0� !�បព<នQម� ��ង�� ម&ន 2ច��5/យ�ផpងA� � ( )2 2

2 2

3 1

2 3

y x y

m x x

x

y y

− + =

+ − =

1. �គ !���� : ABC &ន�ក�Cផ7 S �O/យ ,R r ��@ងA� ' ��ងBង"\� Dក��] ន�ង\� Dក

ក��ង���� : ABC �

E. �គ ! , ,x y z 'ប,- ច�ន�នព��# �ជ%&ន� ��យបI% ក"J�

2 2 3 3

2 2

x y z y x z x y z x y z

y z x z y x x y z x y z

+ + + + + + ++ + ++ + + +

≥+ + +

'()&'()&'()&'()&

ចំេល�យ

�. A 'ច�ន�នប[ម A⇒ 'ច�ន�នគ�"# �ជ%&ន

�យ/ង��e#&ន� 2 3 86 09 ba + − ≥

�យ/ង�ឃ/ញJ� 2 3 86 19 3b ka + − = +

Page 131: Vnmo 30 4-2011-grade 10

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Page 130

29 3 86 3 110 3 10 3.2 103 7a b k kA + − ++ = + = += .

��យ 1(m )27 od13≡ , Sញ;ន 10 3 13.27 0 (mod13)k + ≡ +

Sញ;ន A )ចក�ច"នDង 13

��យ 13'ច�ន�នប[ម �O/យ A 'ច�ន�នប[ម�6� 13A =

�ព5�6� 0k = 2 89 3 6 10ba⇒ + − =

2 2 22 103 9 293a ab a⇒ + = ⇒ ≤ ⇒ ≤ )���យ a 'ច�ន�នគ�"ធមn'��, �យ/ង;ន�

290a b= ⇒ =

261a b= ⇒ =

172a b= ⇒ =

3 2a b= ⇒ = �

�. 2 2

2 2

3 1 (1)

( ) 2 3 (2)

y x y

m y

x

x x y

− + =+ − =

2

2(1)

1

3xy

x +⇒ =

+, ជ�ន�ច*5 (2) �យ/ង;ន�

4( 2) (4 5) 9 0 (*)m m mx − + − + − =

�./ម0� !�បព<នQ&នច��5/យព���ផpងA� (*)⇔ &នU'ច�ន�ន# �ជ%&នម�យ�

ក�:� �� 2 72 / 3m x == ⇒ (យក 2m = )

ក�:� �� 2

0

0

2 142

6

m

m

S

− +

≠∆ = ⇔ =

>

ក�:� 1� 0 2 9P m< ⇔ < <

.*ន�ច� 2 9m≤ < ន�ង 2 142

6m

− +=

1. �យ/ង��យបI% ក";ន 2sin .sin .sin

SR

A B C=

( ). 2sin .sin .sin

(sin sin sin ) sin sin sin

S S A B Cr

R A B C A B C= =

+ + + +

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�យ/ងកL��យបI% ក";ន�

3 3sin sin sin

2N A B C ≤= + +

3 3sin .sin .sin

8M A B C ≤=

Sញ;ន� 2,

2r

S SMR

NM==

�យ/ង;ន 27 2 227 2

2

MR r S

NM

+ = +

9 9 9 2 2

2 2 2

MS

NM M M

= + + +

3 9

4449 2 2 3

.24 4 12 2

MS

NS S

M NM

≥ ≥

E. =ង ; ;b y z cx xa zy= + = + = +

# �មiព 2a b c b c a b c

b c a a b b c⇔ ≥ + + ++ + +

+ +

2 2( )( ) ( )( ) ( )

( ) ( ) ( )( )( )a a b b c b a b b c c a b b c

b c aa b b c a b b c

+ + + + + ++ +⇔ ≥ + + + + + +

2 2

2()2

)(2

a b cb b c

b

c a bc

ca

ab b b

+⇔ ≥ +++ ++

FងR�ង�ឆBង 2 3 2 2 2 3

21 1 1

2 2 2

a b a bc c b c ab

b c b a a

c c

c a c

b = + + + + + + +

4

3 2 22 2 2c

ab ab bc bb

ac ba

≥ + ≥ + ++ +

Iw " "= �ក/�&ន�ព5 a b c= = � M x y z= = �

'()&

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វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២៧

�. �����យម� �� 33 23 2 6 3x xx x− − = −

�. �គ ! n 'ច�ន�នគ�"# �ជ%&ន89ង, ! 3.2125 5 4n n+ − )ចក�ច"នDង 20112 �

��យបI% ក"J n )ចក�ច"នDង 20092 �

1. ក��ងប3ង"�គ !កន3�ប67 �"ប� , ,Ox Oy Oz ��@ងA� ប�ង̀/�'ម�យA� ;នម�� 0120 � =ង A'

ច�ន�ច�2នDង�5/កន3�ប67 �" Ox ( A�ផpងព� O )� =ង B ,C ��@ងA� 'ព��ច�ន�ចច5<��2

�5/កន3�ប67 �"S�ងព�� ,Oy Oz 89ង, ! 2OB OC OA+ = (B ន�ង C �ផpងព� O ) �

�ងBង"\� Dក��]���� : ABC =ម5��ប" �"កន3�ប67 �"ឈមA� �ប"កន3�ប67 �" Ox ,

,Oy Oz ��ង" 1 1 1, ,A B C � ច*�ក�:�"ទ�=�ង�ប" B ន�ង C �./ម0� !ក�នyម�

2 2 21 1 1

21 1 1 1 1 12 2 2OC OA

OA OB OC

OB OC OA OB+ +

+ + +&ន��C5�*ចប�ផ���

E. �គ ! , ,x y z 'ប,- ច�ន�នព��# �ជ%&ន�ផ7GងH7 �"5កq:r 2 2 2y z yx x z+ + = �

��យបI% ក"J� 9 4( )xy yz zx x y z≥ ++ ++ + �

K. �គ ! 100ច�ន�នគ�"# �ជ%&ន, ច�ន�នន�ម�យ�ម�នធ�'ង 100�O/យ&នផ5ប*ក�n/ 200�

��យបI% ក"J ក��ងច��,មច�ន�ន�6� �គbច��ជ/យក;នប9�6n នច�ន�ន).5&ន

ផ5ប*ក�n/នDង 100�

N. ក��ងប3ង" ( )Oxy , �គ !;9 9̂ប*5 2: 2 () 0)( y px pP = > ន�ងប67 �" ( )d �"=មក�ន�� F

�" ( )P ��ង"ព��ច�ន�ច 1M ន�ង 2M � ច*�ក�:�"ទ�=�ង�ប" ( )d �./ម0� !���� :

1 2OM M &នប� �&���*ចប�ផ���

'()&'()&'()&'()&

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ចំេល�យ

�. 33 23 2 6 3x xx x− − = −

3 2 233 4 2 2 2( 3 41 ) 2( 1) x x x xx x⇔ + − − = − − + +−

=ង 23 2( 1) 3 4 2

1

x

x

v x

u

x− − +

= −

+

=

�យ/ង;ន�បព<នQម� �� 3 2

3 2

3 4 2 2 (1)

3 4 2 2 (2)

x x v

v x

u

x u

+ − − =+ − − =

យក (1).ក (2)FងRនDងFងR �យ/ង;ន� 3 3 2 2v uu v− = −

2 2( 2) 0)( uu v u v v+ + +−⇔ =

22 2 2

01 3

2 42 0 2 0

u vu v

u uuv v v v

=− =

+ ⇔ ⇔

+ + + = + + =

�យ/ង;ន 3 2 31 3 6 3 1 0 (*)x x xu v x x⇔ − + ⇔ − − == − =

�ប/ [ ]2; 2x ∈ − , =ង [ ]2cos , 0;x t t π∈=

3 1 2(*) 8cos cos3

26c

90 (

3os 1 )

kt tt t k

π π⇔ − − = ±=⇔ +⇔ ∈= ℤ

)���យ [ ]

2cos9 95 5

0; 2cos9 9

7 72cos

9 9

t x

t t x

t x

π π

π ππ

π π

= =

= = =

=

��យម� � (*) 'ម� �.M��កទ�ប� �6�&ន��ច/នប�ផ��Uប��

.*ច�ន� ម� �&នUប�គM 5 72cos ; 2cos ; 2cos

9 9 9

π π π �

�. �យ/ង&ន� 23.25 4 (1 5 1)(2 25 5 )n n n n+ − = − +

��យ 5 2n + 'ច�ន�ន� �6� 3.2125 5 4n n+ − )ចក�ច"នDង 2011 12 5n⇔ − )ចក�ច"

នDង 20112 �

��យ n 'ច�ន�នគ�"# �ជ%&ន �6�bចប�)បក 2 .kn m= ច��Z� ,k m∈ℕ , m 'ច�ន�ន��

(ម�នម�O��ផ5)

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�យ/ង&ន ( )2 . 25 1 5 151k k m

n m− = − = −

( ) 12 2 2 2 21 (5 ) .5 ( 15 .. 5k k m k km− −− += + + +

��យ m 'ច�ន�ន��6� 2 1 2 2 2) (5 ...(5 5 1k k km m− −+ + + +

'ច�ន�ន�

Sញ;ន 5 1n − )ចក�ច"នDង 2011 2 12 5k

⇔ − )ចក�ច"នDង 20112

�យ/ង;ន 2 12 2 2 21 (5 1)(5 1)(5 1)(5 1) ... (5 5 1)k k−

− = − + + + +

2 12 2 2 2(5 1)(5 1)(5 1) ... (52 1)k−

+ + += +

��យ 2 1 ( 0,1, 2, ..., 1)5i

i k+ = − )ចក�ច"នDង 2 )�)ចកម�ន�ច"នDង 4 , .*ច�6� 25 1k

− )ចក

�ច"នDង 22k+ )�)ចកម�ន�ច"នDង 32k+

.*ច�ន� 25 1k

− )ចក�ច"នDង 2011 22 2k+⇔ )ចក�ច"នDង 20112 �

2011 22 009k k⇔ ≥ ⇔ ≥+ �

Sញ;ន 2 .kn m= )ចក�ច"នDង 20092 �

1. �2�5/កន3�ប67 �" , ,, OyO Ozx =ងប,- #� �ចទ<��ក= 1 2 3, ,e e e� � �

(.*ច�*ប)

=ង I 'ផa���ងBង"\� Dក��]���� : ,ABC M 'ច�ន�ចក,- 5 1AA �

�យ/ង;ន� 1.IA IO OA IO OA e= + = +� � � � �

1 1 1 1.IA IO OA IO OA e= + = −� � � � �

Sញ;ន 1 1 12 ( ).IA IA IO OA OA e+ = + −� � � �

)���យ 1 1 12 2 2 ( ).IA IA IM IM IO OA OA e+ = ⇒ = + −� � � � � �

�យ/ង;ន 1 1 1. 0 . ( ) 0 (12 )IM e IO e OA OA= ⇒ − − =� � � �

.*ចA� ).�, �យ/ង;ន 2 1. ( ) 0 (2)2IO e OB OB+ − =� �

ន�ង 3 1. ( ) 0 (2 3)IO e OC OC+ − =� �

ប*ក (2)(1), , (3)FងRនDងFងR �យ/ង;ន�

1 2 2 1 1 1( ) (2 ) ( ) 0IO e e e OA OB OC OA OB OC+ + + + + − + + =� � � �

��យ 1 2 3 1 1 10 OA OB Oe e e OB OCC OA+ + =+ + = ⇒ + +� � � �

�យ/ង;ន 2 2 21 1 1

1 1 1 1 1 12 2 2OC OA

OA OB OC

OB OC OOA B+ + ++ +

C

A

B

x

y

z

1A

1B

1C

1e�

2e�

3e�

I

M

Page 136: Vnmo 30 4-2011-grade 10

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Page 135

2

1 1 1 1 1 1

1 1 1

(

3( 3

)

)

OB OC OB OC

O

OA O

B C

A

OA O

+ + + +≥+

=+

(=ម# �មiព Schwarz )

2 2 21 1 1

1 1 1 1 1 12 2 2 3

OA OB OC OA OB OCOA

OB OC OA OBOC OA

+ ++ +⇒ ≥+

=+ +

.*ច�ន� 2 2 21 1 1

1 1 1 1 1 1 min2 2 2

OOA OB OC

OB OC OA

OC OA OBA

+ +

=

+ + +

OB OC OA=⇔ = �

E. �យ/ង&ន 2 2 2 22y zxyz yzx x+ >= + ⇒ >

.*ចA� ).� , 22y z> >

�យ/ង;ន 9 4( )xy yz zx x y z≥ ++ ++ +

( 2)( 2) ( 2)( 2) 3 (*( )( ) )2 2x y y z z x− − + − − ≥+ − −⇔ .

=ង 2, 2 ( , , 0)2,b y c za cx a b= − = − >= −

(*) 3ab bc ca+ +⇔ ≥ .

�យ/ង&ន 2 2 2y z yx x z+ + =

2 2 2( 2) ( 2) ( 2)( 2)(2) 2)( b c a b ca⇔ + + + + = + + ++ .

2 2 2 4 2( )b c abc ab bc caa⇔ + + + = + + + .

2 2 2 ( ) 4 4 (**)b c ab bcabc ab bc ca a ca⇔ + + − + + + ≥+ + + = .

=ង 33

3

ab bc ca

aat bc t bc≥ ⇒ ≥+ += (=ម# �មiព Cauchy )

ព� (**) 6� ! 3 2 2( 1)( 2)3 4 0t tt t− ++ ≥ ⇔ ≥ 1 3ab ct bc a⇒ ≥ ⇒ + + ≥

Iw " "= �ក/�&ន 1 3a b c x y z⇔ = = = ⇔ = = = �

K. .�ប*ង �យ/ង�ង̀��ឃ/ញJ 5�l�"ព��'ន�ចa �ប/�គប"ច�ន�ន�ទQ)��n/A�

�k*# �យ/ងព�ន��!ក�:� ក��ង 100 ច�ន�ន 1 2 100, , ...,nn n ).5 !&ន89 ង��ចព��ច�ន�ន�ផpA� ,

zប&J' 1n ន�ង 2n �

ព�ន��! B�����ម&ន 100ច�ន�ន.*ច�ង�� ម� 1 2 1 2 1 2 3 1 2 99, , , , ..., ...n n n nn n n n n n+ + + + + +

+ �ប/&នច�ន�នម�យ ក��ងB����ន� )ចក�ច"នDង 100: ��យច�ន�ន�ន�ធ�'ង 0 �O/យ�*ច'ង

200 �6�?គM�n/នDង 100�

Page 137: Vnmo 30 4-2011-grade 10

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Page 136

+ �ប/ម�ន&នច�ន�ន, ក��ងB����ន�)ចក�ច"នDង 100: �ព5�6� ក��ង 100 ច�ន�ន�ប"B���

�ង�5/, &ន89 ង��ច 2ច�ន�ន&ន�:5".*ចA� �ព5)ចកនDង 100�

.*ច�6� ផ5.ក�ប"ព��ច�ន�ន�ន� )ចក�ច"នDង 100� ��យផ5.ក�ប"ព��ច�ន�ន�ន� ធ�

'ង 0 �O/យ�*ច'ង 200 �6�?�n/នDង 100 (ព��ច�ន�ន�ន� ម�ន)មន' 1n ន�ង 2n )

.*ច�ន� �យ/ង;នបIg ��e#��យបI% ក"�

N. + .�ប*ង �យ/ងព�ន��!ក�:� ( )d )កងនDងF<កp Ox �

�ព5�6� ( )d &នម� � 2

px =

1 2( ), (; ;2 2

)pp p

M p M⇒ −

.*ច�ន� ប� �&������ : 1 2OM M �n/�

1 2 1 2 5(2 )O O M MM pM+ + = +

+ �ប/ ( )d ម�ន)កងនDងF<កp Ox �

=ង k '�មគ�:�;ប"ទ��ប" ( ) ( )2

() : yp

kd xd ⇒ = −

ក*F����នច�ន�ច�បពB�ប" ( )d ន�ង ( )P �ផ7GងH7 �"�បព<នQ�

( )

( )

22

2

2

2

1 1

) 1( 1

22

2

pp k

y xk

p ky

x

py k x

k

− + = −

=⇔

= − + =

2

2 21 2 2 1 2 1 2 1) (

2 1)( 2

p kM x yy

kM px y y⇒ = − + − +≥ = >−

2 2 2 21 2 1 1 2 2OM y yOM x x+ = + ++

2 2 2 2 2 21 1 2 2 2 1 2 1(( ) ) ( )x x xy y x y y+ += + −+− ≥ +

2 2 2 2 2

2 21 2 4 2

4(2 ) (1 )4 5

k kO

p pOM M pp p

k k+ >++ ≥ + =+

6� ! ( )1 2 1 2 2 5OM MO MM p+ + > +

��បមក, �./ម0� !���� : 1 2OM M &នប� �&���*ចប�ផ�� គM ( )d )កងនDងF<កp Ox �

'()&'()&'()&'()&

y

x

d

'M

M

F O

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វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២៨

�. �ក��C5ធ�ប�ផ��, ��C5�*ចប�ផ���ប"ក�នyម� ( ) ( )

( ) ( )10 2 5

2 5 10

1 1

1 1 2

x y x yP

x y y

+ − +=

+ ++,

ច��Z� ,x y ម�នF# �ជ%&ន�

�. �កU'ច�ន�នគ�"�ប"ម� ��ង�� ម� 2 2 2 23 2 1 0y x y xx − − + − =

1. M 'ច�ន�ចម�យ4���2ក��ង���� : ABC � ប,- ប67 �" , ,AM BM CM �" ,BC

,CA AB ��@ងA� ��ង" 1 1 1, ,A B C � �កទ�=�ង�ប" M �./ម0� !ក�នyម

1 1 1

MA MB MCP

MA MB MC= + + &ន��C5�*ចប�ផ���

E. =ង , , , ,a b c r R ��@ងA� 'ប,- �ជmង, ��ងBង"\� Dកក��ង, ��ងBង"\� Dក��]���� : ABC

��យបI% ក"# �មiព� 2 2 2 2( ) ( ) 6 2 )( 3 () b c a b c a b c R ra b c a R++ +− − + + − ≤ − �

K. បfg ញJច�ន�ន 555 2222 55522 + )ចក�ច"នDង 7�

N. �ក ,a b �./ម0� !�បព<នQ�ង�� ម&នច��5/យ)�ម�យគ�"� 2

2 2 2

( )

4

z b I

x

xyz

y

xy

z

z a

z + =

+

+ = + =

'()&'()&'()&'()&

ចំេល�យ

�. ( )( )( ) ( )

5 5

22 5

1

1 1

x y xyP

x y

− −=

+ +

=ង 5 ntan , tayx α β== ច��Z� , 0;2

πα β ∈

( )( )( ) ( )2 2 2 2

2 2

) )tan tan cos cos

cos ) c

sin( cos(.

tan 1 tan cos cos(sin (sin1 tan 1 sin 2 .

cosos )

cos

P

α β α βα β α β α β α β

α α β βα βα β

−= =

+

+

+−

+ +

Page 139: Vnmo 30 4-2011-grade 10

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Page 138

( )( )

1 sin 2 1 1 1 1( )

2 1 sin 2 1 sin 2 2 1 sin 2 1 sin 2

s

2

in 2A B

α βα β β α

= = − = − + + + +

�យ/ង;ន ( )max max min

1

2P A B−= ន�ង ( )min min max

1

2P A B= −

, 0;2

πα β ∈ sin 2 , sin 20 1α β⇒ ≤ ≤

.*ច�ន� max

00

sin 2

sin 2 11

14

04

xP

y

ββπα α

= =

=== ⇒ ⇔

= =

min

1

sin 2 01

sin 2 014

14

xP

y

πβ βα α

=

= == − ⇔ ⇔ ⇔

= = =

�. ( ) ( )22 2 2 2 2 23 2 0 3 (*)11y x y xx y x x− − + − = = −⇔ −

��យ 2y ន�ង 2( 1)x − 'ប,- ច�ន�ន ���;ក. �6� 2 3x − កL'ច�ន�ន ���;ក.).��

.*ច�6� =ង ( )( )2 2 2 23 3 3x x x z x zz z− = ⇔ − = ⇔ − + =

�យ/ង;ន ,x z x z+ − '��)ចក�ប" 3�O/យ x z+ ម�នF# �ជ%&ន �6� x z− កLម�ន

F# �ជ%&ន).�

3 2,

21

1

2, 3

y

x

x z xx

x z y

= ±⇒ ⇒ ⇒

+ = = = − = = − = ±

1. =ង 1 2 1, ,ABM ACM BCMS S S S S S∆ ∆ ∆= = =

�យ/ង;ន� 1 11

ACMABM

BMA CMA

SSAM

MA S S∆∆

∆ ∆

= =

1 1

1 2

3

ABM ACM

BMA CMA

S S

S S

S S

S∆ ∆

∆ ∆

= =+ ++

.*ចA� ).�� 1 3

1 2

SBM

B S

S

M= + ន�ង 2 3

1 1

SCM

C S

S

M= +

1 2 2 3 3 11 2 2 3 3 1

1 2 3 1 2 3

)( )( ) 2 .2 .2(8

S S SS S S S S

S S

S S SSP

S S S S

+ + + ≥= =

មiព�ក/�&ន�ព5 1 1 11 2 3

1 1 1

1 1

3 3ABC

MA MB MCS S

AS

A BB CCS ∆= = = = = =⇔

.*ច�ន� min 8P = �ព5 M 'ទ��បជ��ទ�ងន"���� : ABC �

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 139

E. �យ/ង&ន ( ) ( ) ( )2 2 2a b c a c ab cb a b c+ − + − + −+ +

( ) ( ) ( )4abc b c a c a b a b c= − + − + − + −

( )( ) ( )28

16 8 16S

RS p a p b p c RSp

= − − − − = −

( )8 8 )2 (12S

S R S R rp

= − = −

មu9ង�ទ@� 3 3sin sin sin

2A B C+ + ≤

�O/យ 3sin sin si 3 sin sin in s nA B C A B C≥+ +

3

3 3 3 3sin .sin .sin

6 8A B C

=

⇒ ≤

2 2 3 32 sin sin s . 2)in (

4S R A C RB=⇒ ≤

ព� (1) ន�ង (2) ( ) ( ) ( ) ( )2 2 2 23 26a b c a c a b a b c Rb rc R⇒ + + ≤+ − + − + − −

មiព�ក/�&ន 3sin sin sin

2A B C A B C⇔ ⇔= = = = = , គMJ ABC∆ ម<ងp�

K. ��យ 555 555222 5 (mod 7222 7. ) 5 (mod 7)31 5 222= ⇒ ⇒ ≡+ ≡

មu9ង�ទ@� 2 25 4 (m5 od7)= ≡

3 4.5 7) 6 7)5 (mod (mod≡ ≡

4 6.5 7) 7)5 (mod 2 (mod≡ ≡

5 7)5 2. 3 (mod 7)5 (mod≡ ≡

6 63.5 (mod 7) 1(mo5 5d 7) 1(mod 7)k≡ ≡ ⇒ ≡

)���យ 555 35 (m555 6.9 od 7) 6 (mod 75 )2 3 ⇒ ≡ ≡= +

គMJ 555 6 (m )222 od 7≡

បក��យ.*ចA� ).�, �យ/ង;ន�

222 2222 (mod 7555 7.79 2 555 (mod) 2 7)≡ ⇒ ≡= +

1 2 (m )2 od 7≡

2 4 (m )2 od 7≡

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 140

3 38 (mod 7) 1(mod2 27) 1(mod 7)k≡ ≡ ⇒ ≡

)���យ 22222 1(mod 7)2 3.74 2⇒ ≡=

គMJ 222 1(m )555 od 7≡

.*ច�ន� 555 222222 555 0 (mod 7)+ ≡ គMJ 555 2222 55522 + )ចក�ច"នDង 7�

N. 5កqខ:r \�;ច"�5កqខ:r \�;ច"�5កqខ:r \�;ច"�5កqខ:r \�;ច"� zប&J�បព<នQ ( )I &នU 0 0 0; )( ;x y z , .*ច�6� 0 0 0; ; )( yx z−− កL'

U�ប"�បព<នQ ( )I ).�

��យ�បព<នQ&នច��5/យ)�ម�យគ�"�6� 0 0 0x y= =

ជ�ន�ច*5 ( )I �យ/ង;ន� 0

0

20 4

2

2

za b

a

z

zb

ba

= =

= = −

== ⇔

=

5កqខ:r �គប"�Aន"�5កqខ:r �គប"�Aន"�5កqខ:r �គប"�Aន"�5កqខ:r �គប"�Aន"�

:2a b= = �យ/ង;ន�បព<នQម� �� 2

2 2 2

(1)

2 (2)

4 (3)

2

xyz z

x y

xyz z

z

+ =+ =

+ + =

ព� 2) 1 0( ( ) 2z xyz z+ ⇒=⇒ ≠

យក (1) (2)− FងRនDងFងR �យ/ង;ន� (1 ) 0 (1 ) 0xyz xy zz ⇒ −= =−

2 2

2, 0

0

1

0

0 2,

. 1

3

x

y

z y

x

zy

z

x y

x

= = =

=

⇒ = =⇒ ⇒ =

= ⇒+ =

)��បព<នQ 2 2

5 1

3

2

5 11 2

5 1

2

5 1

2

x

yy

xx

y

y

x

+= − == − = + =

+ =

⇒ �បព<នQ ( )I &នច��5/យប��ផpងA� �

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�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០

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Page 141

2:a b= = − �យ/ង;ន�បព<នQម� �� 2

2 2 2

(4)

2 (5)

4 (6)

2xyz z

xyz z

x y z

+ = −+

+ = −

+ =

ព� ( 1) ) 2(2 0z xyz z⇒ + = ⇒ ≠

យក (4) (5)− FងRនDងFងR �យ/ង;ន� (1 ) 0 (1 ) 0xyz xy zz ⇒ −= =−

2 2

2, 0

0

0

3

2

3

.

, 0

1

z y

y z x

zy

x

x y

x

⇒ = − =⇒ = ⇒ = − =

=

=

= −

⇒+ =

)���យ�បព<នQ 2 2

3

3y

xy

x +−

==

An នU ⇒ �បព<នQ ( )I &នច��5/យ)�ម�យគ�" (0,0, 2)− �

.*ច�ន� �បព<នQ&នច��5/យ)�ម�យគ�"�ព5 2a b= = − �

'()&'()&'()&'()&

វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII

វ��� �ទី២៩

�. �����យ�បព<នQម� ��

2 2

2

2 2

0

2 0

3 8 8 8 2 4 2

y xy yz

x x y yz

x y xy z

x

yz x

+ + + =

+ + + =

+ +

+ − − =

�. �គ ! , ,a b c 'ប,- ច�ន�នព��# �ជ%&ន� ��យបI% ក"# �មiព�ង�� ម�

33 .

3.

2 3

abc a b a b ca

a ab + + +≤+ +

1. �គ !ឆ� : ABCDEF \�Dកក��ង�ងBង"ផa�� O � R &ន AB CD EF R= = = �

=ង , ,H K L '�ជ/ងក�ព").5ទ��ក"ព� O �P�5/ , ,BC DE AF =ម5��ប"�

បfg ញJ ���� : HKL '���� :ម<ងp�

E. �ក�គប"ប,- ច�ន�នគ�" m �./ម0� !ម� � ( )3 2 2 01mx mxx m+ − + =− &នU'

ច�ន�នគ�"�

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K. �គ ! 2011ច�ន�ច�2�5/ប3ង"ម�យ� ��យ.DងJ ក��ង�កmមន�ម�យ�).5&នប�ច�ន�ចក��ង

ច��,5ប,- ច�ន�ច�ង�5/ �ព5,កL&នព��ច�ន�ច).5&ន�ប)#ងខ3�'ង 1�

��យបI% ក"J ក��ងប,- ច�ន�ច�ង�5/ &ន89ង��ច 10064���2ក��ង�ងBង"ម�យ).5

&ន ��n/នDង 1�

N. �គ !�F5�ប ( )E �2នDង&ន O 'ផa��� ម��)កង �xOy # �5ជ��# �ញ O , �ជmងS�ងព�� Ox ន�ង Oy

�ប"? �" ( )E ��@ងA� ��ង" Aន�ង B � ��យបI% ក"J AB )�ងប9�នDង�ងBង"នDងម�យ�

'()&'()&'()&'()&

ចំេល�យ

�. �បព<នQមម*5នDង 2 2 2 2

(*)

( 1) (2 1) 0

4( ) 4( ) ( 1) (2 1)

( ) ( ) 0

x x y x

x y y z x

z

x

x x y y y

+ + + =+ + + = + + +

+ + + =

ព�ន��!�ម/5ប,- #� �ចទ<� ( ; ), ( ; ), ( 1; 2 1)a x y b x y y z c x z+ + + +�� �

�បព<នQ 3 យ�P' 2 2

. 0 (1)

. 0 (2)

(3)4

a b

a c

c b

=

=

=

��

� �

��

+ �ប/ 0a =��

�6� 0x y= = �O/យព�ម� �ទ� 3�ប"�បព<នQ �យ/ង;ន 1

2z = −

+ �ប/ 0a ≠�� �6� b

� &នទ��d.*ច c

�ព5�6� (3) c b⇒ = ±��

)a �ប/ 2c b=�� �6�

1 2( )

2 1 2( )

( ) ( ) 0

x x y

z y z

x x y y y z

+ = + + = + + + + =

0x⇔ = ន�ង 1

2y = �O/យ 1

2z = −

)b �ប/ 2c b= −�� �6�

1 2( )

2 1 2( )

( ) ( ) 0

x x y

z y z

x x y y y z

+ = − + + = − + + + + =

(=ម (*) )

(=ម (*) )

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Page 143

2 5 2

3 1

2 2

5 04

:

3

y x

z x

x x

+

= − − =

+ =

ន���h ន� �បព<នQ&នច��5/យ 1 1 10; , 0; ;

2 2;

20 − −

�. 3 3. .2

a bab ab abab

+≤=

�យ/ង��យបI% ក"J 33 3

2 2(1)

2

xyzx y z =≥+ +

�យ/ង&ន� 3

2 312 3

.3

a aa a a b a b c

a b a b c

+ ++ +

+≤ +

+ +

3

31 13

3

bb a b c

a b c

+ ++≤ +

+ +

3

2 312 3

.3

b cb c a b a b c

a b a b c

+ ++ +

+≤ +

+ +

ប*កFងRនDងFងR �យ/ង;ន 3 33 32 3.

23.

a babca ab

a b a ba

c

++ + + + + ≤

.*ច�ន� (1) ��e#;ន��យបI% ក"� Sញ;ន # �មiព�./មព���

1. =ង � �BOH HOC α= =

� �DOK KOE β= =

� �FOL LOA γ+ =

2 3 23

2 2 .πα β γ π+ + + =

�យ/ង;ន 2

πα β γ+ + =

ក��ង���� : OHK �យ/ង;ន�

2 2 2 2 . .cos( )3

OH OK OH OKHKπα β= + − + +

B

A

C

D

E

F

L

H

K

(ម� �An នU)

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Page 144

2 2 2(cos cos 2cos cos cos( ))6

rπα β α β γ+ + +=

2 2 2 2 2(cos cos cos 2cos cos co 2coss cos c cos sin sin os6 6

rπ πα β γ α β γ α β γ γ= −+ + −+

)� 2 2cos sin sin cos cos cos sin 1 sin6

2cosπα β γ γ α β γ γ+ = + −

(cos cos sin ) 1 sin (1 sin cos cos cos( ))γ α β γ γ α β α β− = + − += + .

si1 ns sn ini α β γ= + .

.*ច�ន� 2 2 2 2 2(cos cos cos 2cos cos cos cos6

rHKπα β γ α β γ= + + + − sinsin sin 1)α β γ −

ក�នyម�ន� ឆ3��A� ច��Z� , ,α β γ &នន<យJ ម�នb�<យនDង HK �

ព��6� �យ/ង;ន� HK KL LH= = � M���� : HKL '���� :ម<ងp�

E. zប&J p 'ច�ន�នគ�").5 3 2 2( 1) 0mp mp p m− + − + =

�ព5�6� 2 )( 1( )mp p m+ − = � ��យ p ន�ង m 'ច�ន�នគ�"�6�&នព��ក�:� �

• ក�:� ទ��� 2 1m p mp + = − = −

Sញ;ន 1m p= + �6� 2 1 1pp + + = − (An នU)

• ក�:� ទ��� 2 1m p mp + = − =

Sញ;ន 1m p= − �6� 2 1 1p p+ − = Sញ;ន 2;1p = −

.*ច�ន� 3;0m = − �

K. =ង A 'ច�ន�ចម�យក��ងច��6ម 2011ច�ន�ច).5 ! � ង"�ងBង" ( )A ផa�� A ��n/ 1�

�ប/�គប" 2010ច�ន�ច�25" �ទQ)�4���2ក��ង ( )A �6�5�l�"��e#;ន�����យ��ច�

zប&J B 'ច�ន�ច�2��] ( )A , �ព5�6� 1AB > � ង"�ងBង" ( )B ផa�� B &ន ��n/ 1,

=ង C 'ច�ន�ចទ�ប�ក��ង 2009ច�ន�ច�25"� ព�ន��!�កmម).5&នប�ច�ន�ច ,,A B C �ង�5/�

�យ/ង&ន 1AB > , �6�=មប�^ប", �យ/ង��e#&ន 1AC < � M 1BC < &នន<យJ C 4���2

ក��ង ( )A � M C 4���2ក��ង ( )B , Sញ;ន �ងBង"S�ងព�� ( )A ន�ង ( )B &នផ7�ក�គប" 2011ច�ន�ច

).5 !�

�ង̀�� 2011 2.1005 1= +

.*ច�ន� =ម# �oន Dirichlet &ន�ងBង"ម�យក��ងច��6ម�ងBង"S�ងព�� &នផ7�ក89ង��ច 1006

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ច�ន�ច �

N. ��ជ/�� /����យ ).5&ន O �ធB/'គ5"����យ , Ox 4���2�5/F<កpធ��ប" ( )E , Oy 4���2

�5/F<កp�*ច�ប" ( )E � =ង 2a ន�ង 2b ��@ងA� '�ប)#ងF<កpធ� ន�ងF<កp�*ច�ប" ( )E ,

ម� � ( )E គM� 2 2

2 21

x y

a b+ =

=ង 0Ax By+ = 'ម� ��ប" OA �6�ម� � OB គM� 0Bx Ay− =

ក*F����នច�ន�ច A 'U�ប"�បព<នQ

2 2 2 2 2 2

2 22 22 2 2 2 2 2 2 2

2 2

0;

1

b A b By

Ax Bya a

xx ya a

a bA b B A b B

+ =

⇒ = =+ +

+ =

.*ច�ន� 2 2 2 2

2 2 22 2 2 2

( )A A

a bO

A Bx y

AA

b Ba

+= + =+

.*ចA� 2 2 2 2

22 2 2 2

( )b A Ba

a B AOB

b

+=+

=ង h 'ក�ព"���� : OAB �6� h កL'�ប)#ងព� O �P AB ).� �យ/ង;ន�

2 2

2 2 2 2 2 2 2

1 1 1const

a ab

h OA OB a

bh

b a b

+⇒ ==

+= + =

ប67 �" AB ច5<� &ន�ប)#ង�Pច�ន�ច O �2នDង 'ច�ន�ន�ថ� h , .*ច�ន� AB )�ងប9��PនDង

�ងBង"នDងម�យ &នផa�� O ន�ង ��n/ h �

'()&'()&'()&'()&

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