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Ñîôèéñêè Óíèâåðñèòåò Ñâ. Êëèìåíò Îõðèäñêè Ïèñìåí êîíêóðñåí èçïèò ïî ìàòåìàòèêà, 26 þëè 2007ã. ÒÅÌÀ 3

Çàäà÷à 1. Ðåøåòå óðàâíåíèåòî Çàäà÷à 2. Ëèöåòî íà ðîìá ðàâåí íà

60 . 13

5−x = 2. log2 √ 1−x

ABCD

å ðàâíî íà

120,

à ðàäèóñúò íà âïèñàíàòà â íåãî îêðúæíîñò å

Íàìåðåòå äúëæèíèòå íà äèàãîíàëèòå

Çàäà÷à 3. Íàìåðåòå âñè÷êè ðåøåíèÿ íà óðàâíåíèåòî Çàäà÷à 4. Îêîëî îñòðîúãúëåí òðèúãúëíèê ñ ðàäèóñ

1.

ABC

AC

è

BD.

x = 4. 2 √ BC = 3 å

sin2 x + 6 sin2

ñúñ ñòðàíà

îïèñàíà îêðúæíîñò

Îò âúðõà

A å ñïóñíàò ïåðïåíäèêóëÿð êúì äîïèðàòåëíàòà êúì îêðúæC , êîéòî ïðåñè÷à äîïèðàòåëíàòà â òî÷êà M è CM = 1. Íàìåðåòå ∢ACB .

íîñòòà ïðåç âúðõà ãîëåìèíàòà íà

Çàäà÷à 5. Íàìåðåòå âñè÷êè ñòîéíîñòè íà ðåàëíèÿ ïàðàìåòúð

−9 ≤ ñà èçïúëíåíè çà âñÿêî ðåàëíî ÷èñëî

p,

ïðè êîèòî íåðàâåíñòâàòà

3x2 + px − 6 ≤6 x2 − x + 1

x.

ABC ñà BC = 4 è AB = 9. ÍàAC , àêî å äàäåíî, ÷å ∆ABC å ïîäîáåí íà òðèúãúëíèê ðàâíè íà âèñî÷èíèòå íà ∆ABC .

Çàäà÷à 6. Íàé-ìàëêàòà è íàé-ãîëÿìàòà ñòðàíè â òðèúãúëíèêà ìåðåòå äúëæèíàòà íà ñòðàíàòà ñ äúëæèíè íà ñòðàíèòå,

Çàäà÷à 7. Íåêà

n

å åñòåñòâåíî ÷èñëî. Äîêàæåòå, ÷å çà âñÿêî ÷èñëî

ñòâîòî

(x + 1)n +

³1

x

x>0

å èçïúëíåíî íåðàâåí-

´n + 1 ≥ 2n+1 .

Çàäà÷à 8.  òðèúãúëíèêà

ABC äúëæèíèòå íà ìåäèàíàòà CM îò âúðõà C (M ∈ AB ) è úãëîïîëîâÿùàòà BL íà ∢ABC (L ∈ AC ) ñå îòíàñÿò êàêòî 5 : 6, îêîëî ÷åòèðèúãúëíèêà M BCL ìîæå äà ñå îïèøå îêðúæíîñò è AB = 18. Íàìåðåòå äúëæèíèòå íà ñòðàíèòå AC è BC .

Çàäà÷à 9. Íåêà

G å ìåäèöåíòúð íà ïðàâîúãúëåí òðèúãúëíèê ABC âúçìîæíî íàé-ãîëÿìàòà ñòîéíîñò íà cotg ∢AGB .

ñ õèïîòåíóçà

AB . Íàìåðåòå

f (x) = x2 + ax + b è g(x) = x2 − ax + c, êúäåòî ðåàëíèòå ÷èñëà a, b è c óäîâëåòâîðÿâàò íåðàâåíñòâîòî 2a2 (b + c) + (b − c)2 < 0. Äîêàæåòå, ÷å âñÿêî îò óðàâíåíèÿòà f (x) = 0 è g(x) = 0 èìà ðåàëíè è ðàçëè÷íè êîðåíè.

Çàäà÷à 10. Äàäåíè ñà ôóíêöèèòå

Âðåìå çà ðàáîòà - 5 ÷àñà Äðàãè êàíäèäàò-ñòóäåíòè,

íîìåðèðàéòå âñè÷êè ñòðàíèöè íà áåëîâàòà ñè;

ðåøåíèåòî íà âñÿêà çàäà÷à òðÿáâà äà çàïî÷âà íà íîâà ñòðàíèöà;

÷åðíîâàòà íå ñå ïðîâåðÿâà è íå ñå îöåíÿâà.

Èçïèòíàòà êîìèñèÿ âè ïîæåëàâà óñïåøíà ðàáîòà!


!"#$%&# '(#)*+%#,*, - ). /0#1*(, 23+#4%&#5 6#%1*( &!(&7+%*( #89#, 9! 1:,*1:,#&: ;< =0# ;>>?@. ABCBDE E 6FEGHFIE FHJHIEK L MHGB N A:4:O: P. !"!#! $%&'(!()!#* log2 √51−−xx = 2+

!"!#$!% ,!-)().)*((&#& */0&1# ! x ∈ (−∞, 1)+ ,&2!(*#* $%&'(!()! ! !3')'&0!(#(* (& √5 − x = 44 ) 5%) 1−x x ∈ (−∞, 1) 06'&#& 7$ 1#%&(& ! 5*0*8)#!0(&+ 90!2 5*'2):&(! (& 2'!#! 1#%&() ' 3'&2%&# ) *1'*/*82&'&(! *# ;(&7!(&#!0 5*0$<&'&7! $%&'(!()!#* x2 + 6x + 9 = 04 ) *# #$3 x1 = x2 = −3+ =>? 3&#* x = −3 5%)(&20!8) (& 2!-)().)*((&#& */0&1#4 #* ! %!"!()! (& 2&2!(*#* $%&'(!()!+

A:4:O: ;. @).!#* (& %*7/ ABCD ! %&'(* (& 1204 & %&2)$1># (& '5)1&(&#& ' (!:* *3%>8(*1# ! %&'!( (& 60 + 13 A&7!%!#! 2>08)()#! (& 2)&:*(&0)#! AC ) BD+

!"!#$!% B3* a ! 2>08)(&#& (& 1#%&(&#& (& %*7/&4 & h ! ')1*<)(&#& 7$4 )7&7! S = a.h = a.2r4 #+!+ 120 = a. 120 4 13 *#3>2!#* a = 13+ A!3& AC = x ) BD = y Cx > 0, y > 0D4 #*:&'& 1& );5>0(!() %&'!(1#'&#& AC.BD = 2S ) AC 2 + BD2 = 4a2 4 #+!+ xy = 240 ) x2 + y2 = 676+ !"!()6#& (& #&;) 1)1#!7& 1& x1 = 24, y1 = 10 ) x2 = 10, y2 = 24+ 90!2*'&#!0(* ;& 2)&:*(&0)#! (& %*7/& )7&7! 2'! '>;7*8(*1#)E AC = 24 ) BD = 10 )0) AC = 10 ) BD = 24+

A:4:O: N. A&7!%!#! '1)<3) %!"!()6 (& $%&'(!()!#* sin2 x + 6 sin2 x2 = 4+

!"!#$!% F&7!1#'&7! ' 2&2!(*#* $%&'(!()! sin2 x = 1 − cos2 x ) 2 sin2 x2 = 1 − cos x ) 5*0$<&'&7! !3')'&0!(G #(*#* $%&'(!()! cos2 x + 3 cos x = 04 #+!+ (cos x + 3) cos x = 0+ H# #$3 cos x + 3 = 0 )0) cos x = 0+ I>%'*#* *# #!;) $%&'(!()6 (67& %!"!()!4 & %!"!()6#& (& '#*%*#* $%&'(!()! 1& x = π + kπ4 k G .60* <)10*+ 2

A:4:O: Q. H3*0* *1#%*>:>0!( #%)>:>0()3 ABC 1>1 1#%&(& BC

= 3 ! *5)1&(& *3%>8(*1# 1 %&2)$1 1+ H# '>%J& A ! 15$1(&# 5!%5!(2)3$06% 3>7 2*5)%&#!0(&#& 3>7 *3%>8(*1##& 5%!; '>%J& C 4 3*?#* 5%!1)<& 2*5)%&#!0(&#& ' #*<3& M ) CM = 1+ A&7!%!#! :*0!7)(&#& (& ∢ACB + √

!"!#$!% H# 1)($1*'&#& #!*%!7& ;& △ABC )7&7! BC = 2 sin α = 34 *#3>2!#* (&7)%&7! sin α = 3 + I* 2 $10*')! △ABC ! *1#%*>:>0!(4 ;&#*'& α = 60o + K7&7! *L! ∢ACM = ∢ABC = β C);7!%'&# 1! 1 !2(& ) 1>L& 2>:& *# *3%>8(*1##&D+ H# 5%&'*>:>0()6 #%)>:>0()3 ACM ) 1)($1*'&#& #!*%!7& 5*0$<&'&7! CM = AC. cos β = 2 sin β cos β = sin 2β = 1+ =>? 3&#* β ! >:>0 ' #%)>:>0()34 10!2'&4 <! β = 45o + H# #$3 (&7)%&7! ∢ACB = 180o − α − β = 180o − 60o − 45o = 75o .

A:4:O: R. A&7!%!#! '1)<3) 1#*?(*1#) (& %!&0()6 5&%&7!#>% p4 5%) 3*)#* (!%&'!(1#'&#& −9 ≤

3x2 + px − 6 ≤6 x2 − x + 1

1& );5>0(!() ;& '163* %!&0(* <)10* x+ !"!#$!% F&/!06;'&7!4 <! x2 − x + 1 > 0 ;& '163* x4 ;&#*'& 2'!#! (!%&'!(1#'& 1& !3')'&0!(#() (& 1)1#!7&#& *# 3'&2%&#() (!%&'!(1#'& 12x2 + (p − 9)x + 3 ≥ 0 ) 3x2 − (p + 6)x + 12 ≥ 0+ F& 2& />2! );5>0(!(* '163* *# #!;) (!%&'!(1#'& ;& '163* x4 ! (!*/J*2)7* ) 2*1#&#><(* 2)13%)7)(&(#)#! (& 1>*#'!#1#'&L)#! )7 3'&2%&#() $%&'(!()6 2& 1& (!5*0*8)#!0()4 #+!+4 D1 = (p − 9)2 − 144 ≤ 0 ) D2 = (p + 6)2 − 144 ≤ 0+ !"!()6#& (& D1 ≤ 0 1& p ∈ [−3, 21]4 & (& D2 ≤ 0 1& p ∈ [−18, 6]+ 9!<!()!#* (& #!;) 2'& )(#!%'&0& () 2&'& #>%1!()#! 1#*?(*1#) (& 5&%&7!#>%&E p ∈ [−3, 6]+

A:4:O: <. A&?G7&03&#& ) (&?G:*067&#& 1#%&() ' #%)>:>0()3& ABC 1& BC = 4 ) AB = 9+ A&7!%!#! 2>08)G

(&#& (& 1#%&(&#& AC 4 &3* ! 2&2!(*4 <! ∆ABC ! 5*2*/!( (& #%)>:>0()3 1 2>08)() (& 1#%&()#!4 %&'() (& ')1*<)()#! (& ∆ABC + !"!#$!% ,& *;(&<)7 BC = a4 AC = b4 AB = c4 ) (!3& ha 4 hb ) hc 1& 1>*#'!#()#! ')1*<)() 3>7 #6J+ 9>:0&1(* $10*')!#* (& ;&2&<&#& )7&7! a ≤ b ≤ c4 *#3>2!#* 5%!2')2 aha = bhb = chc (= 2S△ABC ) ! );5>0(!(* ha ≥ hb ≥ hc + =*:&'& 2'*?3)#! 1>*#'!#() 1#%&() ' 2'&#& 5*2*/() #%)>:>0()3& )7&# 2>08)() a ) hc 4 b ) hb 4 c ) ha 4 ) ;&#*'& a : b : c = hc : hb : ha + M)82&7!4 <! #%6/'& 2& ! );5>0(!(* hc : hb = a : b4 & *#√2%$:& 1#%&(& )7&7! hc : hb = b : c+ H# #$3 ;&30N<&'&7!4 <! a : b = b : c4 #+!+ b2 = ac+ =&3& 5*0$<&'&7! AC = 4.9 = 6+


!"!#! $% !"# n ! !$%!$%&!'( )*$+(, -("#.!%!/ )! 0# &$1"( )*$+( x > 0 ! *023+'!'( '!4#&!'$%&(%( (x + 1)n +

´n + 1 ≥ 2n+1 .

x ´n ³ ³1 1 ´ + 1 = (x + 1)n 1 + n , f (x) = (x + 1)n + x x

!"!#$!% 5#06+!.7#8! 9:'";*1%# (0, ∞)

³1

* 24(*0&(7'#%# *= !

<1 ! 7*9!4!';*4:!8# &

³ (x + 1)n n(x + 1)n−1 n+1 n(x + 1)n−1 1 ´ (x − 1) = (x − 1)(1 + x + x2 + · · · + xn ). f ′ (x) = n(x + 1)n−1 1 + n − n n+1 = n+1 x x x xn+1

>*.7#8!/ )! f ′ (x) < 0 24* x ∈ (0, 1) * f ′ (x) > 0 24* x ∈ (1, ∞), ?+!7(&#%!+'( f (x) ! '#8#+1&#@# & *'%!4&#+# (0, 1) * 4#$%1@# & (1, ∞), A% %:" 0#"+B)#&#8!/ )! 24* x > 0 ! *023+'!'( f (x) ≥ f (1) = 2n + 2n = 2n+1 ,

!"!#! &% > %4*363+'*"# ABC 73+.*'*%! '# 8!7*#'#%# CM (% &34C# C DM ∈ AB E * 36+(2(+(&1@#%# BL '# ∢ABC DL ∈ AC E $! (%'#$1% "#"%( 5 : 6/ ("(+( )!%*4*363+'*"# M BCL 8(.! 7# $! (2*F! ("43.'($% * AB = 18, #8!4!%! 73+.*'*%! '# $%4#'*%! AC * BC , !"!#$!% !"# 7# &3&!7!8 (0'#)!'*1%# BC = a * AC = b, A% %(&#/ )! BL ! 36+(2(+(&1@# '# ∢ABC * ("(+( )!%*4*363+'*"# M BCL 8(.! 7# $! (2*F! ("43.'($% $+!7&#/ )! ∢ABL = ∢LBC = ∢LCM = ∢LM C , A% %:" 0#"+B)#&#8!/ )! %4*363+'*;*%! AM C * ALB $# 2(7(G'*, A% $&(H$%&(%( '# 36+(2(+(&1@#%# & %4*363+'*"# *8#8! AL : LC = AB : BC = 18 : a, A% %:" * (% AL + LC = b '#8*4#8! AL = a 18b , A% △AM C ∼ △ALB 2(+:)#&#8! + 18 AM AC CM = = , AL AB BL

%,!,/

9 18b a+18

=

5 b = . 18 6

A% 2($+!7'*%! 7&! :4#&'!'*1 '#8*4#8! b = 15 * a = 7/ %,!, AC = 15 * BC = 7, !"!#! '% !"# G ! 8!7*;!'%34 '# 24#&(363+!' %4*363+'*" ABC $ C*2(%!':0# AB , #8!4!%! &308(.'( '#HI6(+18#%# $%(H'($% '# cotg ∢AGB , !"!#$!% !"# BC = a/ AC = b/ ∢AGB = ϕ/ * '!"# A1 * B1 $# $4!7*%! $3(%&!%'( '# BC * AC , A% $&(H$%&(%( '# 8!7*;!'%34# 0'#!8/ )! AG = 23 AA1 * BG = 32 BB1 , A% %:" * (% J*%#6(4(&#%# %!(4!8# 0# ¢ 1 ¢ 1 4¡ 4¡ △AA1 C * △BB1 C '#8*4#8! AG2 = AC 2 + CA21 = (4b2 + a2 )/ BG2 = BC 2 + CB12 = (4a2 + b2 ), A% 9 9 9 9 "($*':$(&#%# %!(4!8# 0# △ABG 2(+:)#&#8! ¢ ¢ 1¡ 2 2 4¡ 2 4b +a +4a2 +b2 −9a2 −9b2 = − a2 +b2 ⇒ AG.BG cos ϕ = − (a2 +b2 ). 9 9 9

2AG.BG cos ϕ = AG2 +BG2 −AB 2 =

A% 2S△AGB = AG.BG sin ϕ * S△AGB = 31 S△ABC = 16 ab $+!7&# AG.BG sin ϕ = 31 ab, 2 + b2 , A% 2(+:)!'*%! *04#0* 0# AG.BG sin ϕ * AG.BG cos ϕ '#8*4#8! cotg ϕ = − 32 a ab 4 A% '!4#&!'$%&(%( a2 + b2 ≥ 2ab 2(+:)#&#8! cotg ϕ ≤ − 32 2ab = − / "#%( 4#&!'I ab 3 $%&(%( $! 7($%*6# $#8( 24* a = b, K %#"#/ 8#"$*8#+'#%# $%(H'($% '# cotg ∢AGB ! − 43 * $! 7($%*6# 24* 4#&'(G!74!' 24#&(363+!' %4*363+'*", !"!#! ()% -#7!'* $# 9:'";**%!

f (x) = x2 + ax + b * g(x) = x2 − ax + c/ "37!%( 4!#+'*%! )*$+# a/ b 2a2 (b + c) + (b − c)2 < 0, -("#.!%!/ )! &$1"( (% :4#&'!'*1%# f (x) = 0 *

* c :7(&+!%&(41&#% '!4#&!'$%&(%( g(x) = 0 *8# 4!#+'* * 4#0+*)'* "(4!'*, !"!#$!% L! *02(+0&#8! $+!7'*1 *0&!$%!' 9#"%M #"( 0# "&#74#%'#%# 9:'";*1 h(x) $ 2(+(.*%!+!' "(!9*;*I !'% 24!7 x2 $3@!$%&:&# 4!#+'( )*$+( x0 / %#"(&# )! h(x0 ) < 0/ %(6#&# :4#&'!'*!%( h(x) = 0 *8# 7&# 4#0+*)'* 4!#+'* "(4!'#, !4#&!'$%&(%( & :$+(&*!%( '# 0#7#)#%# '! 8(.! 7# G37! *023+'!'( 24* a = 0/ 0#%(&# a 6= 0 * (% %:" f (x) 6= g(x), L! 2("#.!8/ )! 64#9*"*%! '# f (x) * g(x) *8#% !7*'$%&!'# 24!$!)'# %()"# (x0 , y0 ), #*$%*'#/ −b , ? '!2($4!7$%&!'( 24!$81%#'! 2(+:)#&#8! :4#&'!'*!%( f (x) = g(x) 24*%!.#&# !7*'$%&!'( 4!F!'*! x0 = c 2a ¤ 1 £ 2 y0 = f (x0 ) = g(x0 ) = 2 2a (b + c) + (b − c)2 < 0, J4*+#6#8! 6(4'(%( %&347!'*! $ h(x) = f (x) * h(x) = g(x)/ 4a 0# 7# 0#"+B)*8/ )! &$1"( (% :4#&'!'*1%# f (x) = 0 * g(x) = 0 *8# 7&# 4!#+'* * 4#0+*)'* "(4!'#, &'(#)*) +!"!#$! #, -./0, 1,2,3, .! )4!#/-, . 5 *)30$% 64!#0,*, .! 7)(83,-, 7) 9)+:8(,*, 2 + 0, 1.N ; 0'2!*) N ! <+)/* #, 7)(83!#$*! *)30$%

2007.26.07 Софийски университет "Св. Климент Охридски"  
2007.26.07 Софийски университет "Св. Климент Охридски"