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�បជំុវ�� គណិតវ�ទ�សិស�ពូែកគណិតវ�ទ��� ក់ទី១០
បកែ�បេ�យ ែកវ សិរ"
គណិតវ�ទ�អូ�ំពិក�បៃពណីេវៀត�ម 30-4 ��ំ 2011
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 1
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��� '()&'()&'()&'()&
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 2
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 =
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 3
� 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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 4
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⇔ + + + + ≥
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 5
+ 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 − + − =∆+
=
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 6
.*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+ + = �
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 7
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 �
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ព��'.*ច�ន�, ព�ន��!ចtប"ប�)5ង\�ងផa�� ( '): ( )V O O→ � �ព5�6� XY d→
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�ប"ធ�* XY �
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=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
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⇒ ∃ចtប"ប�)5ង\�ង ប�)5ង DEF∆ �P' 1 1 1A B C∆
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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
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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+ + + + ≥−
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+ 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+ =+ ≥
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+ .*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
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k k n kn
k
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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− − −
= = =− + − −= −∑ ∑ ∑
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(1 ) (1 )n n
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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
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11
1
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2
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nk k n kn
k
n nk k n k k k n kn n
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k
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k
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k C x x
nx n
n
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n C x x nx n n
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− −−
=
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= =
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=
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∑
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() 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
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x xf x ≤ =+ −
, Iw មiព�ក/�&ន�ព5 1
2x =
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4f x = ទទ�5;ន�ព5 1
2x = �
'()&'()&'()&'()&
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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 �
'()&'()&'()&'()&
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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= + ≥
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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
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1
)1
0 1 2(
(1 23
1(1 2 1 2 0
) 0
3
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uu
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uu
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u
u
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30 4 2011 34 17
x y
x yx y
+ = ⇔ = = −
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10
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a
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c
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�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⇒ ⇒= + == +−
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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⇔ = + .
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�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
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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 =
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" ក��ង 5ច�ន�នគ�",កL��យ, )�ង&នប�ច�ន�ន).5&នផ5ប*ក)ចក�ច"នDង 3" �
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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���
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)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")�
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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�
'()&
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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
⇔ + + − + =
⇔ + +=
− − =⇔
=
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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= + − −− + + +
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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)� 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 = − + − + − + + + + + +
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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
= = = = ∩ ∩ ∩ ∩
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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") �
'()&'()&'()&'()&
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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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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 ≡+
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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+ + + ≡ .
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.*ច�ន� ក�:� �ន�ម�ន�ផ7GងH7 �"�
+ ក�:� �: ក��ងព��ច�ន�ន ,p q &នម�យ)ចក�ច"នDង 3
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.*ច�ន� {2; 3; 5; 7}q∈ (��Z� q 'ច�ន�នប[ម)
ច��Z� 2q = �6� 18 10 3) 4(n n n= + ⇒ =+
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ច��Z� 7q = �6� 18 70 3) 8(n n n= + ⇒ =+
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2 ; 7 ; 3 ; 3
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q q q q
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= = = = = = = = = = = =
�
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ង"���� :)កងម;� APQ ( ;AP QPQ= ន�ង B 4���2�ង)�ម�យ�ធ@បនDង AP ) ,
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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��ចព�ន��!ចtប"បងB�5 045
AQ (ផa�� ;A ម���ងB�5 045 ) : 'P P֏ ន�ង 'B B֏
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�យ/ង;ន 2PQ AP= = ន�ង 2 ' 'QC P B= 2 3 2PB= =
��យ 2 3 2PQ QC CP PC≤ + ⇒ ≤ +
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2 2 24 (2 3 2) 26 12 2AC PCAP = + + = +⇔ = + �
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1
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9 92
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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 = )
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+ + + + =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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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 = �
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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
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=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
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24 9 5 3 171;
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វ�1��េស��រ�បឡងអូពំិចេវ�ត�ម*� ក់ទី១០ េល�កទី XVII
វ��� �ទី៥
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(3 )( 3 ) 14
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x y x y
x x y
xy
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+ =
�. �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 �
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E. �គ ! , ,a b c 'ប,- ច�ន�នព��# �ជ%&ន� បfg ញJ� 23
2 2 228
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≥+ +
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 ) � '()&'()&'()&'()&
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ចំេល�យ
�. �����យ�បព<នQម� � 2 2
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10 3(2)
15 15
3 14(1
36),
uv u
u
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=
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6 5 4 2 3 3 2 4 5 6
6 5 4 2 3 3 2 4 5 6
6 6
66
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6 15 20 15 6
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=
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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 =
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មu9ង�ទ@� 10 1(mod112 ) (4)≡
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�6� ( )220 102 1(m 1)2 od1= ≡ , Sញ;ន 212 (m2 )od11≡ , ផ7�យនDង (3)�
.*ច�ន� 2p = ��e#�\5�
+ ព�ន��! 3p =
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Sញ;ន 322 1( 3)mod3≡/ ន�ង 322 2 ( 3)mod3≡/
.*ច�ន� 3p = ��e#�\5�
+ ព�ន��! 11p =
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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 =
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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
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A A A B A Ca
AA = =2
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.*ចA� ).� �យ/ង;ន� 2 3
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Iw �n/�ក/�&ន ABC⇔ ∆ '���� :ម<ងp�
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កL=ម# �មiពក*��, �យ/ង&ន� 3( )
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ព� (*) Sញ;ន � M � 060OAB > � M � 060OBA > �
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.*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 �
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
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).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
= =
⇒
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Uទ�ព��គM 23(17;6)
32
y
n
==
⇒
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+ ក�:� ទ��: 2 22 1 2 7 07 1x y m y my m⇒ + − −+ = =+
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
==
⇒
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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
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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
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��យ�ប)#ង�ប"�3 ប�n/ 1
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�*ប�ជ/ង)ក�កទ�ម�យ�
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ព�ន��!�ម/5�ងBង").5&ន ��n/ 5�ក=� �ក�Cផ7�ងBង" 25π , �ងBង"ន�ម�យ�).5&ន
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2 2&ន�ក�Cផ7 1
8π �
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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�
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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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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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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 =
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�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
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)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
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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�
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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( )( )( ) ( )( )( )( ) ( )( )
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 = �
'()&'()&'()&'()&
�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⇔ + − = ⇔ + − =
�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
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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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.*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"
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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= −
+
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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��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/�
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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=
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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 ).��
�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"�
'()&
�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
++
≤ ≤ �
'()&'()&'()&'()&
�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�
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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�
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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���=)��
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
�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"
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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⇔ + + − + > .
�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)
�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
�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 )
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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( )
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⊥ � '()&'()&'()&'()&
�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���
�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
�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�
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 77
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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 78
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
+ + += + +− − − − − −
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 79
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 =
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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�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��
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�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"?�
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+ �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
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c
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ប*ក��មនDង (*) Sញ;ន#� �ចទ<� 1 1u OB OC= +� �� &នទ�.*ចA� នDង OA
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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
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�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���
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,).5o3 ប"�2ក��ង�ទmង'ម�យA� ព��ព5ម�ន 4���2ក��ង�ទmង'ម�យA� ម-ង�ទ@��ទ�
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N. �ក k �./ម0� !�បព<នQ�ង�� ម&នច��5/យ��ច/នប�ផ��� 2 2
1 (1)| 1|
(
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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
− + − =−
+ =
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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
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= −= = − =
�
.*ច�ន� �បព<នQ).5 !&នច��5/យ 0 2( 1; ; )− −
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�. ��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�"
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π �
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+ − ≤<
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) tani ivvπ π
+ − ≤< < �
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��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+ +
++ +
< = < − −+
−−+
−=
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)b �ប/ 1 17017 16
v vπ ππ+ − ≤< <
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) n(6
tav vv vππ+ − << =−
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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
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A
AA
=+
, ជ�ន� tan ; sin2 2
A r aA
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−
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2
2
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p
ra p a
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a r RR
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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 −− + + =+
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�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− − − − �
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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− �
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�;ក.' ម�ន&ន&ន"ព��កt5,�ក"'ន"A� �ទ ��Z�ប,- )ខpធ�* ន�ង ��ង�5/�ទQ)�
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N. �).ង�2�5/ប3ង" ( )Oxy , ប,- ច�ន�ច ( , )M x y �ផ7GងH7 �" (1)' �� ABCD ).5&ន
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'�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"
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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
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�"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 +
+ = − =
+
=
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732
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u
w v v
wu w
− = = = − = =
⇔=
+ +
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5 1 1
x y x
x y y
+ = = − − = =
⇔
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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 �
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.*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⇔ + + =
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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 −=
�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
�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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
−− + =
=
�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⇒ ∈ − − − − − − − − −− �
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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)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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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.*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≥
+ + + + + ++ + =
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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+ = ⇔ + − + =
+ + �
'()&
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 99
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 =≥+ +
�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
�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
+ +=
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 102
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 �"
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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Page 103
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
−+ − = =
⇔ ⇔ ⇔ = =− + = − − + =
�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= + = + + ++ ∈� � � � � � � �
ℝ
.*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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
+ + + + + + + + + + + + + + + ++
≤ +
�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
�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+ + + = �
�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
�
�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
�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)
�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
�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 )�
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
− + + += +⇔
= + = +
=
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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(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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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=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�
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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(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+ + = ⇒ + + =� � � � �� �
.
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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
�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⇔ + + + =
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
+ + + =+ − + =
�����យ.*ច )a �យ/ង;នច��5/យព�� (1; 12)− ន�ង ( 29; 1 )− − �
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∆
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
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)( )( ( ( ( () ) ))
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
+ + + +=
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+ + + ++ + ++ ++ +
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4 91
b c
ab
ab bc
bc ca ab bc ca
a aa cc b≥
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++ +
++ +
2 2 2 2
1 4 9
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+ + ++ + + + +⇔ ≥
+
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+ + ++ + + +++
+ (ព��)
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K. ច�ន�ន�ន���ង &នច�ន�នo��'ច�ន�ន� ��e#)�'ច�ន�នគ* ��Z� 2n 'ច�ន�នគ*, zប&J
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ប��"ទ�o�� �យ/ងនDង;ន�P.5"ក�:� ).5 �គប"�ន���ងS�ងF"�ទQ)�&នច�ន�នo��
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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
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2 2 2 211 2
2
APHPH
PPM PA A
PA MPP= ⇔ =
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222 2 2 2 2
2H
by a x a
ax⇔ − −=
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( )2 2 2 2 2H Hy a a xb⇔ = − (��Z� Hx x= )
2 2
42
2
1H Hx y
aab
⇔ + =
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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
− + =
+ − =
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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
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�យ/ង��e#&ន� 2 3 86 09 ba + − ≥
�យ/ង�ឃ/ញJ� 2 3 86 19 3b ka + − = +
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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⇒ + − =
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290a b= ⇒ =
261a b= ⇒ =
172a b= ⇒ =
3 2a b= ⇒ = �
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2 2
3 1 (1)
( ) 2 3 (2)
y x y
m y
x
x x y
− + =+ − =
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2(1)
1
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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− − = −
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��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 �
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(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
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− = − + + + +
2 12 2 2 2(5 1)(5 1)(5 1) ... (52 1)k−
+ + += +
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i k+ = − )ចក�ច"នDង 2 )�)ចកម�ន�ច"នDង 4 , .*ច�6� 25 1k
− )ចក
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.*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
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1C
1e�
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I
M
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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�
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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�
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+ �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
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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
−= =
+
−
+
+−
+ +
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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( )( )
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 �
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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≡
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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� �
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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�"�
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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 (*) )
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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)
�បជុំវ���គណតិវ�ទ��សិស��ពូែកគណតិវ�ទ�����កទ់ី១០
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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�
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