2007.20.05 Софийски университет "Св. Климент Охридски"

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-!,!#! #,!#$% *+!' '( .,/0,!"$%#(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(!)'/ " .,!)./+/@!'$!#/ x ≥ 0S'$ )(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 $'#!,2(+( [0, π2 ]3 !"!#$!% D $'#!,2(+( [0, π ] E9'6F$$#! sin x $ cos x .,$!-(# '!/#,$F(#!+'$ "#/G'/"#$: 7(#/2( -/@!- )( ./2)$0'!- )2!#! "#,('$ '(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 = π2 3 ;/2( "( $ 2"$*6$#! ,!&!'$% '( )()!'/#/ 9,(2'!'$! 2 $'#!,2(+( [0, π2 ]3 ?:4:L: O. H$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+$7(K$ /# !)$' /# 21,I/2!#! '( .(,(+!+!.$.!)(3 8# 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'!'$%: '(-$,(-! a2 b2 c2 = 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-(#( M A + M C 3 !"!#$!% 8*!2$)'/ M '! #,%J2( )( +!@$ 21,I9 )$(0/'(+( AC : $ -/@!- )( .,!)./+/@$-: *! M ∈ ∆ACD3 O6/ M ! 21#,!&'( 7( ∆ACD: '!6( N ! .,!"!*'(#( #/*6( '( .,(2(#( AM "1" "#,('(#( CD3 8# '!,(2!'"#2/#/ M C < M N + N C ./+9*(2(-!

AM + M C < AM + M N + N C = AN + N C : #3!3: AM + M C -/@! )( "! 92!+$*$: (6/ $7J!,!- M ≡ N 3 8# ),90( "#,('(: (6/ N 6= D: #/0(2( /# AN < AD + DN "+!)2( AN + N C < AD + DN + N C = AD + DC = b + a3 86/'*(#!+'/ ./+9*(2(-!

AM +M C ≤ a+b: 6(#/ ,(2!'"#2/#/ ! $7.1+'!'/ "(-/ .,$ M ≡ D $+$ .,$ M ≡ B 3 ;(6( 217-/@'/ '(G50/+%-(#( "#/G'/"# '( M A + M C ! 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(@!#!: *! M N = AN + M B 3 !"!#$!% ( /7'(*$- AC = BC = a: AM = y: M B = x $ CM = z 3 8# 9"+/2$!#/ '( 7()(*(#( "+!)2(: *! ∢ABC = ∢BAC = 45o : ( /"2!' #/2( $-(-! ∢AM C = ∢ABC $ ∢BM C = ∢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( √ 2 2 2 2 2 2 2 ∆AM C $ ∆BM C '(-$,(-! y + z − 2yz 2 = z + x − 2xz 2 = a : /#61)!#/ √ √ √ ./+9*(2(-! x2 − 2xz = y2 − 2yz: (x − y)(x + y) = 2z(x − y): $ ./'!@! x 6= y: x+y =

√

2z. √

;,$101+'$61# M N C ! .,(2/101+!' $ ,(2'/J!),!': √ 7(#/2( √ z = 2M N 3 P(#/ 7(-!"#$- 2 0/,'/#/ ,(2!'"#2/: ./+9*(2(-! x + y = 2. 2M N = 2M N : #3!3: M B + AN + N M = 2N M : /#61)!#/ "+!)2( M B + AN = M N 3

!"!#! $% !" #$%&'(#)*+ ) %,-+.)+ 4 + 1 )" -#/+%,* !&(0(# ! *#1$, M 2 , #34,*, +5 !&(0(, -#/+%,*"6(, )" -#/+%, -# *78 ! *#1$+ A + B 9 :,5"%"*" 6+;"*# (, *%+&<&6(+$, AM B 9 !"!#$!% :"$, O1 + O2 ), ;"(*%#!"*" (, -!"*" #$%&'(#)*+ ) %,-+.)+ )&#*!"*(# 4 + 12 , P " /"*,*, (, /"%/"(= -+$.67%,2 )/.)(,* #* M $&5 AB 9 >%"? *#1$,*, O2 /%"$,%!,5" /%,!,2 .)/#%"-(, (, AB 2 $#7*# /%")+1, #*)"1= $+*" M P + O1 A )&#*!"*(# ! *#1$+*" N + Q9 @%+&<&6(+$&* O1 O2 Q " /%,!#&<&6"( ) $,*"* O1 Q = 4 − 1 = 3 + 8+/#*"(.?, O1 O2 = 4 + 1 = 52 + #* >+*,<#%#!,*, *"#%"5, (,5+%,5" O2 Q = 49 @&A $,*# QO2 BA " /%,!#&= <&6(+$2 )6"-!,2 1" AB = 49 @%+&<&6(+;+*" N M O2 + QO1 O2 ), /#-#3(+2 ?,= *#!, OM NQ = M3N = OO2OM = 15 2 #*$&-"*# (,5+%,5" M N = 35 9 B6"-#!,*"6(# 1 2 1 ?, !+)#1+(,*, $&5 )*%,(,*, AB ! ∆AM B /#6.1,!,5" M P = M N + N P = 8 1 16 3 + 1 = 9 C+;"*# S (, ∆AM B (,5+%,5" #* S = AB.M P = 9 5 5 2 5 !"!#! &% D *%+&<&6(, /+%,5+-, ABCD %&3#!"*" AD2 BD + CD +5,* %,!(+ -&6'+(+ + ), -!, /# -!, !?,+5(# /"%/"(-+$.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.√ #)(#!,*, √ ABC 2 *#<,!, H " ;"(*&%&* (, #/+),(,*, #$#6# ∆ABC #$%&'(#)*2 , %,-+.)&* √ 3 +G " AH = 2ℓ 3 = ℓ 36 9 H"(*&%&* (, #/+),(,*, #$#6# /+%,5+-,*, )F"%, 6"'+ !&%8. /%,!,*, DH 2 + ("$, #?(,1+5 ) M -%.<+7 $%,A (, -+,5"*&%, (, )F"%,*,2 $#A*# 5+(,!, /%"? *#1$,*, D9 @%+&= <&6(+$&* AM D " !/+),( ! #$%&'(#)* ) -+,5"*&% AM = 2R2 )6"-#!,*"6(# " /%,!#&<&6"( ) /%,! &<&6 /%+ !&%8, A2 + AH " !+)#1+(, $&5 8+/#*"(.?,*, 5. DM 9 I* >+*,<#%#!,= √ p 2 2 *, *"#%"5, ?, /%,!#&<&6(+7 *%+&<&6(+$ AHD (,5+%,5" DH = AD − AH = ℓ 33 9 I* √

F#%5.6,*, AD = DH.DM ?, /%,!#&<&6(+7 *%+&<&6(+$ DAM /#6.1,!,5" ℓ = ℓ 33 .2R2 + #* *.$ ℓ = √2 R9 @&A $,*# #$#6(+*" %&3#!" DA2 DB + DC ), -!, /# -!, !?,+5(# /"%= 3 /"(-+$.67%(+2 AD " !+)#1+(, $&5 )*"(,*, BCD (, /+%,5+-,*, ABCD2 + ?, #3"5, + GV √ 1 BD.CD 1 3 4 3 3 1 (,5+%,5" V = 3 SBCD .AD = 3 . 2 .AD = 6 ℓ = 27 R 9 !"!#! '% 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#(+*" ! $.*+7*,2 *#<,!, /&%!#*# -"*" " !?"6# 1 + (s − 1)/10 ¡ ¢ 3#(3#(,2 , !*#%#*# " !?"6# 2 + s − 3 − (s − 1)/10 /10 3#(3#(,9 ># .)6#!+"2 *" ), !?"6+ %,!"( 3%#A 3#(3#(+2 + #* /%+%,!(7!,("*# (, -!,*, +?%,?, /#6.1,!,5" s = 819 I* *.$ (,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$, Q !&%8. )*%,(,*, AB 2 *,$+!, 1" P C + QB = BC 9 >%"? *#1$+*" P + Q " /%"$,%,(, #$%&'(#)*2 $#7*# )" -#/+%, -# )*%,(,*, BC ! *#1$, 1 M + ∢QM P = 90o − ∢BAC 9 :,5"%"*" -&6'+(,*, (, CM 2 ,$# P C = 29 2 !"!#$!% , #?(,1+5 ) α2 β + γ &<6+*" (, ∆ABC )&#*!"*(# /%+ !&%8#!"*" A2 B + C 9 :"$, *#1$,*, M1 !&%8. )*%,(,*, BC " *,$,!,2 1" BM1 = BQ + CM1 = CP 9 @#<,!, *%+&<&6(+;+*" P CM1 + QBM1 ), %,!(#3"-%"(+2 + 2

∢P M1 Q = 180o − ∢CM1 P − ∢BM1 Q = 180o −

2

180o − γ 180o − β α − = 90o − .

2 2 2

:# /# .)6#!+" ∢P M Q = 90o − α2 2 + /#("'" M " -#/+%(,*, *#1$, (, #$%&'(#)**, -# )*%,(,*, BC 2 %,!"()*!#*# ∢P M1 Q = ∢P M Q 4" 3&-" +?/&6("(# ),5# ,$# M1 ≡ M N! /%#*+!"( )6.1,A *#1$,*, M1 " !&(0(, ?, #$%&'(#)**,2 + *#<,!, ∢P M1 Q < ∢P M QO9 B6"-#!,*"6(# CM = CP = 29 &'(#)*) +!"!#$! #, -./0, 1,2,3, .! )4!#/-, . 5 *)30$% 64!#0,*, .! 7)(83,-, 7) 9)+:8(,*, 2 + 0, 1.N ; 0'2!*) N ! <+)/* #, 7)(83!#$*! *)30$%