!"#$%&# '(#)*+%#,*, - ). /0#1*(, 23+#4%&#5

6#%1*( &!(&7+%*( #89#, 9! 1:,*1:,#&:

;< 1:$ ;<<=>.

?@A@BC C 6DCEFDGC DFHFGCI J KFE@ ;

?:4:L: M. !"!#$%# $ &!"#'()!"!#$%# *+!' '( (,$#-!#$*'( .,/0,!"$% "( "1/#2!#'/ ,(2'$ '( 14 $ 263 4(5-!,!#! #,!#$% *+!' '( .,/0,!"$%#(3

!"!#$!% 4!6( a1 $ d "( "1/#2!#'/ .1,2$%# *+!' $ ,(7+$6(#( '( (,$#-!#$*'(#( .,/0,!"$%3 8# 9,(2'!'$%#(a10 = a1 + 9d = 14 $ a16 = a1 + 15d = 26 '(-$,(-! d = 2: a1 = −43 ;/0(2( a3 = a1 + 2d = −4 + 4 = 03

?:4:L: ;. <!&!#! '!,(2!'"#2/#/ 3|x| ≤ 2 + |x + 1|3 !"!#$!% ()!'/#/ '!,(2!'"#2/ ! !62$2(+!'#'/ '( '!,(2!'"#2(#(=

1) − 3x ≤ 1 − x, (6/ x < −1; 2) − 3x ≤ 3 + x, (6/ − 1 ≤ x < 0; 3) 3x ≤ 3 + x, (6/ x ≥ 0.

<!&!'$!#/ '( >? ! x ≥ −1/2: 6/!#/ ! '!"12-!"#$-/ " .,!)./+/@!'$!#/ x < −1A ,!&!'$!#/ '( B? ! x ≥ −3/4:6/!#/ 7(!)'/ " −1 ≤ x < 0 '$ )(2( ,!&!'$! x ∈ [−3/4, 0)A '(6,(%: ,!&!'$!#/ '( C? ! x ≤ 3/2: 6/!#/ 7(!)'/" .,!)./+/@!'$!#/ x ≥ 0 '$ )(2( ,!&!'$! x ∈ [0, 3/2]3 86/'*(#!+'/: 7( ,!&!'$%#( '( )()!'/#/ '!,(2!'"#2/./+9*(2(-! x ∈ [−3/4, 0)

⋃

[0, 3/2] = [−3/4, 3/2]3

?:4:L: N. 4(-!,!#! 2"$*6$ ,!&!'$% '( 9,(2'!'$!#/ sin x + cos x = 1: 6/$#/ "( 2 $'#!,2(+( [0,π

2]3

!"!#$!% D $'#!,2(+( [0,π

2] E9'6F$$#! sin x $ cos x .,$!-(# '!/#,$F(#!+'$ "#/G'/"#$: 7(#/2( -/@!- )(

./2)$0'!- )2!#! "#,('$ '( )()!'/#/ 9,(2'!'$! '( 62(),(#: .,$ 6/!#/ ./+9*(2(-! !62$2(+!'#'/#/ 9,(2'!'$!

2 sin x cos x + sin2 x + cos2 x = 1, #3!3 1 + sin 2x = 1,

$+$ sin 2x = 03 ;1G 6(#/ 2x .,$'()+!@$ '( $'#!,2(+( [0, π]: !)$'"#2!'$#! ,!&!'$% '( ./"+!)'/#/ 9,(2'!'$!

"( x1 = 0 $ x2 =π

23 ;/2( "( $ 2"$*6$#! ,!&!'$% '( )()!'/#/ 9,(2'!'$! 2 $'#!,2(+( [0,

π

2]3

?:4:L: O. H$F!#/ '( /6/+'(#( ./21,I'$'( '( .,(2/101+!' .(,(+!+!.$.!) ! ,(2'/ '( 10: 16 $+$ 18 2 7(2$5"$-/"# /# #/2( 6/% /# "#!'$#! -9 ! $7J,('( 7( /"'/2(3 4(-!,!#! /J!-( '( .(,(+!+!.$.!)(3

!"!#$!% ( /7'(*$- " a, b $ c )1+@$'$#! '( ,1J/2!#!: $7+$7(K$ /# !)$' /# 21,I/2!#! '( .(,(+!+!.$.!)(38# 9"+/2$!#/ ./+9*(2(-! "$"#!-(#( 9,(2'!'$% 2a(b+c) = 10, 2b(a+c) = 16, 2c(a+b) = 18: $+$: !62$2(+!'#'/:ab+ac = 5: ab+bc = 8 $ ac+bc = 93 L1J$,(G6$ ./*+!''/ #!7$ #,$ 9,(2'!'$%: ./+9*(2(-! 2(ab+bc+ca) = 22:$+$ ab + bc + ca = 113 M,!7 ./*+!''/ $72(@)('! /# ./"+!)'/#/ 9,(2'!'$! '(-$,(-! bc = 6: ca = 3 $ab = 23 N-'/@(2(G6$ ./*+!''/ ./"+!)'$#! #,$ 9,(2'!'$%: '(-$,(-! a2b2c2 = 36: /#61)!#/ 7( /J!-( V '(

.(,(+!+!.$.!)( '(-$,(-! V = abc = 63

?:4:L: P. ;/*6(#( M ! 212 21#,!&'/"##( $+$ ./ 6/'#9,( '( .,(2/101+'$6 ABCD "1" "#,('$ AB = a $BC = b3 4(-!,!#! 217-/@'/ '(G50/+%-(#( "#/G'/"# '( "9-(#( MA + MC3

!"!#$!% 8*!2$)'/ M '! #,%J2( )( +!@$ 21,I9 )$(0/'(+( AC: $ -/@!- )( .,!)./+/@$-:

*! M ∈ ∆ACD3 O6/ M ! 21#,!&'( 7( ∆ACD: '!6( N ! .,!"!*'(#( #/*6( '(

.,(2(#( AM "1" "#,('(#( CD3 8# '!,(2!'"#2/#/ MC < MN + NC ./+9*(2(-!

AM + MC < AM + MN + NC = AN + NC: #3!3: AM + MC -/@! )( "! 92!+$*$:

(6/ $7J!,!- M ≡ N 3 8# ),90( "#,('(: (6/ N 6= D: #/0(2( /# AN < AD + DN"+!)2( AN +NC < AD +DN +NC = AD +DC = b+a3 86/'*(#!+'/ ./+9*(2(-!AM +MC ≤ a+b: 6(#/ ,(2!'"#2/#/ ! $7.1+'!'/ "(-/ .,$M ≡ D $+$ .,$M ≡ B3;(6( 217-/@'/ '(G50/+%-(#( "#/G'/"# '( MA + MC ! a + b3

?:4:L: Q. 86/+/ ,(2'/J!),!' .,(2/101+!' #,$101+'$6 ABC " .,(2 101+ .,$ 21,I( C ! /.$"('( /6,1@5

'/"#3 D1,I9 )10(#(

⌢

AB /# /6,1@'/"##(: 6/%#/ '! "1)1,@( #/*6(#( C: ! 27!#( #/*6( M 3 8# #/*6(#( C !

".9"'(# .!,.!')$69+%, 61- .,(2(#( AM : 6/G#/ .,!"$*( /#"!*6(#( AM 212 21#,!&'( #/*6( N 3 /6(@!#!:*! MN = AN + MB3

!"!#$!% ( /7'(*$- AC = BC = a: AM = y: MB = x $ CM = z3 8# 9"+/2$!#/ '( 7()(*(#( "+!)2(: *!∢ABC = ∢BAC = 45o

: ( /"2!' #/2( $-(-! ∢AMC = ∢ABC $ ∢BMC = ∢BAC: #1G 6(#/ "1/#2!#'$#! )2/G6$10+$ "! $7-!,2(# " !)'$ $ "1K$ )10$ /# /6,1@'/"##(3 8#J!+%72(-!: *! ∢CAM! /"#1,: 7(K/#/ 2 .,/#$2!' "+9*(G #/*6(#( N '%-( )( ! 21#,!&'( 7( /#"!*6(#(

AM 3 P(#/ "+!)"#2$!: ∢BAM < 45o: $ y > x3 8# 6/"$'9"/2(#( #!/,!-( 7(

∆AMC $ ∆BMC '(-$,(-! y2 + z2 − 2yz√

2

2= z2 + x2 − 2xz

√

2

2= a2

: /#61)!#/

./+9*(2(-! x2−√

2xz = y2−√

2yz: (x−y)(x+y) =√

2z(x−y): $ ./'!@! x 6= y:

x + y =√

2z.

;,$101+'$61# MNC ! .,(2/101+!' $ ,(2'/J!),!': 7(#/2( z =√

2MN 3 P(#/7(-!"#$- 2 0/,'/#/ ,(2!'"#2/: ./+9*(2(-! x + y =

√2.√

2MN = 2MN : #3!3:MB + AN + NM = 2NM : /#61)!#/ "+!)2( MB + AN = MN 3

!"!#! $% !" #$%&'(#)*+ ) %,-+.)+ 4 + 1 )" -#/+%,* !&(0(# ! *#1$, M 2 , #34,*, +5 !&(0(, -#/+%,*"6(,

)" -#/+%, -# *78 ! *#1$+ A + B9 :,5"%"*" 6+;"*# (, *%+&<&6(+$, AMB9

!"!#$!% :"$, O1 + O2 ), ;"(*%#!"*" (, -!"*" #$%&'(#)*+ ) %,-+.)+ )&#*!"*(# 4 + 12 , P " /"*,*, (, /"%/"(=-+$.67%,2 )/.)(,* #* M $&5 AB9 >%"? *#1$,*, O2 /%"$,%!,5" /%,!,2 .)/#%"-(, (, AB2 $#7*# /%")+1, #*)"1=$+*" MP + O1A )&#*!"*(# ! *#1$+*" N + Q9 @%+&<&6(+$&* O1O2Q "

/%,!#&<&6"( ) $,*"* O1Q = 4 − 1 = 3 + 8+/#*"(.?, O1O2 = 4 + 1 = 52 +#* >+*,<#%#!,*, *"#%"5, (,5+%,5" O2Q = 49 @&A $,*# QO2BA " /%,!#&=

<&6(+$2 )6"-!,2 1" AB = 49 @%+&<&6(+;+*" NMO2 + QO1O2 ), /#-#3(+2 ?,=

*#!,

MN

O1Q=

MN

3=

O2M

O2O1

=1

52 #*$&-"*# (,5+%,5" MN =

3

59 B6"-#!,*"6(#

?, !+)#1+(,*, $&5 )*%,(,*, AB ! ∆AMB /#6.1,!,5" MP = MN + NP =3

5+ 1 =

8

59 C+;"*# S (, ∆AMB (,5+%,5" #* S =

1

2AB.MP =

16

59

!"!#! &% D *%+&<&6(, /+%,5+-, ABCD %&3#!"*" AD2 BD + CD +5,* %,!(+ -&6'+(+ + ), -!, /# -!,

!?,+5(# /"%/"(-+$.67%(+9 E,-+.)&* (, #/+),(,*, #$#6# /+%,5+-,*, )F"%, " R9 :,5"%"*" #3"5, (, /+%,5+=-,*,9

!"!#$!% , #?(,1+5 ) ℓ -&6'+(,*, (, #$#6(+*" %&3#!" DA2 DB + DC (, /+%,5+-,*,2 *#<,!, #)(#!,*,

G+ ABC " %,!(#)*%,("( *%+&<&6(+$ ) -&6'+(, (, )*%,(,*, ℓ√

29 :"$, H " #%*#<#(,6(,*, /%#"$;+7 (,

!&%8, D !&%8. #)(#!,*, ABC2 *#<,!, H " ;"(*&%&* (, #/+),(,*, #$#6# ∆ABC #$%&'(#)*2 , %,-+.)&*

G+ " AH =√

2ℓ

√3

3= ℓ

√6

39 H"(*&%&* (, #/+),(,*, #$#6# /+%,5+-,*, )F"%, 6"'+ !&%8. /%,!,*, DH2 + ("$,

#?(,1+5 ) M -%.<+7 $%,A (, -+,5"*&%, (, )F"%,*,2 $#A*# 5+(,!, /%"? *#1$,*, D9 @%+&=<&6(+$&* AMD " !/+),( ! #$%&'(#)* ) -+,5"*&% AM = 2R2 )6"-#!,*"6(# " /%,!#&<&6"() /%,! &<&6 /%+ !&%8, A2 + AH " !+)#1+(, $&5 8+/#*"(.?,*, 5. DM 9 I* >+*,<#%#!,=

*, *"#%"5, ?, /%,!#&<&6(+7 *%+&<&6(+$ AHD (,5+%,5" DH =√

AD2 − AH2 = ℓ

√3

39 I*

F#%5.6,*, AD2 = DH.DM ?, /%,!#&<&6(+7 *%+&<&6(+$DAM /#6.1,!,5" ℓ2 = ℓ

√3

3.2R2

+ #* *.$ ℓ =2√3R9 @&A $,*# #$#6(+*" %&3#!" DA2 DB + DC ), -!, /# -!, !?,+5(# /"%=

/"(-+$.67%(+2 AD " !+)#1+(, $&5 )*"(,*, BCD (, /+%,5+-,*, ABCD2 + ?, #3"5, G+ V

(,5+%,5" V =1

3SBCD.AD =

1

3.BD.CD

2.AD =

1

6ℓ3 =

4√

3

27R39

!"!#! '% J%./, #* n -";, )+ %,?-"6+6+ $.*+7 3#(3#(+9 >&%!#*# -"*" !?"6# 1 3#(3#( + #4" "-(, -")"*, #*#)*,(,6+*" ! $.*+7*, 3#(3#(+9 B6"- ("<# !*#%#*# !?"6# 2 3#(3#(, + #4" "-(, -")"*, #* #)*,(,6+*" )6"- *#!,3#(3#(+ ! $.*+7*,2 + *9(92 /%"-/#)6"-(#*# -"*" !?"6# n − 1 3#(3#(, + #4" "-(, -")"*, #* #)*,(,6+*" )6"-*#!, 3#(3#(+ ! $.*+7*,9 K, /#)6"-(#*# -"*" ! $.*+7*, #)*,(,6+ n 3#(3#(,9 :,5"%"*" 3%#7 n (, -";,*,2 ,$#" +?!")*(#2 1" /&%!+*" -!" -";, ), !?"6+ /# %,!"( 3%#A 3#(3#(+9

!"!#$!% :"$, s " /&%!#(,1,6(+7* 3%#A (, 3#(3#(+*" ! $.*+7*,2 *#<,!, /&%!#*# -"*" " !?"6# 1 + (s − 1)/103#(3#(,2 , !*#%#*# " !?"6# 2 +

(

s − 3 − (s − 1)/10)

/10 3#(3#(,9 ># .)6#!+"2 *" ), !?"6+ %,!"( 3%#A 3#(3#(+2+ #* /%+%,!(7!,("*# (, -!,*, +?%,?, /#6.1,!,5" s = 819 I* *.$ (,5+%,5"2 1" /&%!+*" -!" -";, ), !?"6+ /#9 3#(3#(,9 L" -#$,'"5 1%"? +(-.$;+72 1" !)7$# #* /M%!+*" n − 1 -";, " !?"6# 9 3#(3#(,9 @#!, " !7%(# ?,/&%!+*" -!" -";,2 + ("$, -#/.)("52 1" /&%!+*" m -";, N2 ≤ m < n−1O ), !?"6+ /# 9 3#(3#(,2 *#<,!, )&<6,)(#.)6#!+"*#2 (m+1)=!#*# -"*" " !?"6# m+1+

(

81− 9m− (m+1))

/10 = 9 3#(3#(,9 B6"-#!,*"6(# /&%!+*" n− 1-";, ), !?"6+ /# 9 3#(3#(,2 +6+ #34# 9(n−1) 3#(3#(,2 $#+*# -#3,!"(+ $&5 n=*" 3#(3#(,2 !?"*+ #* /#)6"-(#*#-"*"2 /%,!7* 81 3#(3#(,2 *9"92 9(n − 1) + n = 819 I* *.$ (,5+%,5" n = 99

!"!#! ()% D *%+&<&6(+$, ABC ), !?"*+ *#1$, P !&%8. )*%,(,*, AC + *#1$, Q !&%8. )*%,(,*, AB2 *,$+!,1" PC + QB = BC9 >%"? *#1$+*" P + Q " /%"$,%,(, #$%&'(#)*2 $#7*# )" -#/+%, -# )*%,(,*, BC ! *#1$,

M + ∢QMP = 90o − 1

2∢BAC9 :,5"%"*" -&6'+(,*, (, CM 2 ,$# PC = 29

!"!#$!% , #?(,1+5 ) α2 β + γ &<6+*" (, ∆ABC )&#*!"*(# /%+ !&%8#!"*" A2 B + C9 :"$, *#1$,*, M1 !&%8.

)*%,(,*, BC " *,$,!,2 1" BM1 = BQ + CM1 = CP 9 @#<,!, *%+&<&6(+;+*" PCM1 +

QBM1 ), %,!(#3"-%"(+2 +

∢PM1Q = 180o − ∢CM1P − ∢BM1Q = 180o − 180o − γ

2− 180o − β

2= 90o − α

2.

:# /# .)6#!+" ∢PMQ = 90o − α

22 + /#("'" M " -#/+%(,*, *#1$, (, #$%&'(#)**, -#

)*%,(,*, BC2 %,!"()*!#*# ∢PM1Q = ∢PMQ 4" 3&-" +?/&6("(# ),5# ,$# M1 ≡ M N!

/%#*+!"( )6.1,A *#1$,*, M1 " !&(0(, ?, #$%&'(#)**,2 + *#<,!, ∢PM1Q < ∢PMQO9B6"-#!,*"6(# CM = CP = 29

&'(#)*) +!"!#$! #, -./0, 1,2,3, .! )4!#/-, . 5 *)30$% 64!#0,*, .! 7)(83,-, 7) 9)+:8(,*, 2+0, 1.N ;0'2!*) N ! <+)/* #, 7)(83!#$*! *)30$%