Resenja okruzno 2007

Page 1

ABG BGC CGF AGF AF E EF D CF D CDE ADE ACF ABC ABF BCF ACE ACD

!" # $#% & ' ( ' ) * ' ! $ $ ! ! $ $ ! $ $ $ $ $ ! !

' + ' ,- +

' + ' $ . ' , x / / 0 y m 1 , ' Æ 0 & 2 0 3 Æ 8x m / , ' Æ 0 &

,- 0 4 ' x = 2 5 ' ' , m2 4·2·2+2·2·16+2·2·y. 1 80 + 4 · y = 176 ' y = 24 ! 6 ' ' 7 ' ' 0 $

3 · 565 %8 9 ' ' ' 0 $ 2007 − 1695 2 & + ' ' 0 1 + '

, % ' 0 ! 8 7 9 ' : + Æ : / & ' : = 6 * , - -

0 7 ' 0 2 0 7 0 2 '

' 5 = 9 ; = 0 9 , - - 0 7 0 0 7 ' ( = 10 + < ' ( = 7 ( = 8 . / ( = 8 ' < = 6 0 7 ' & ,& 1 ( = 7 ! < = 4 9 ' 9 + = = 10 > +1 = 9 & 0 7 ' ' 9 = 2 ! = = 8 > = 1 . ' 712 + 59708 = 60420


5 5 a 15 34 = 15 12 = 10 20 10 < 10 < 20 a = 6 a = 7 a + 15b = 56

10 a = 7 b = 2

10 m = 100 dm 1 000 000 ! ! " # 1 m Æ

$ ! ! % & Æ $ 100 000 ! ! ' % $ 100 000 dm 10 km " ( ) *+ ) $ ",- ) ..- / 2p 4r - 3q 0 q 1 q = 2 - 2 2p + 4r = 2000 p + 2r = 1000 2r p p = 2 - 2 r = 499 3 4.. - 5 p = 2 q = 2 r = 499 4 0 ! 5 '6 4) n n '- 4 . 286 = 2 · 11 · 13 429 = 3 · 11 · 13 $ 7 n = 11

n = 13 n = 143 Æ $ 7 ) ( Æ 5 ) n + 1 ) n = 11 $ 7

) " 1 ! C F

E G

A

D

B


ab = 19

a = 3 b = 27 OBM ODN ! " # $

BD OB = OD

$ BOM = DON = 90◦ % M BO ∼ = N DO % M B = N D & D N C # '

" M BN D & ·O · (

Æ A M B ) * ) a b ( + a + b , + a − b

(a + b) · (a − b) - . * - , ) a b /

a − b - )

" a + b - ) "

(a + b) · (a − b) - ) - )

. - . 0 F " ADE BED DE ,

ABC AB # F DE = AF D AF D = ADF 1 ! AF D C AF = AD . ! " D $ D E AC AF = AC 2 . /

BF = BC 2 % AB = A B F AC+BC 1 2 & 23 &41 5 566 0 a '

b ' cm , a + 2b a 7 '

'


a < 2b a + 2b = 2005 a < 1003 a {1, 3, . . . , 1001}


√ √ 17 > 4 37 > 6 √ √ √ √ √ 17 + 37 > 10 10 > 3 2 > 1 √ √ √ √ 17 + 37 + 2 > 5 + 3 + 1 ! " " # 5 + "

$ ! % % " " & % A1 A2 , . . . A12 % " % A B C D " ' ( &! " " ' " & " x & cm $ ! % M # " ! " % A6 " " '& CD ) & % & & A6 M A7 & A6 A7 M " 30â—¦ * +, & # " " & - A8 A7 M C D √ 3 x A6 M = 2 A7 M = 2 x A9 A6 . / & A6 CM A10 A5 " & O M C = x2 A11 A4 A8 D √= A7 C CD = x + A12 A3 2 · ( 23 x + x2 ) √ 0 A A1 A2 B 3) = 10 x · (2 + √ x = 10 · (2 − 3) 1 Æ % 2 " " " 32 > 23 0 " , % # % 32000 > 23000 0 32007 > 37 · 23000 3 32007 − 23000 > (37 − 1) · 23000 37 − 1 > 2007 (37 − 1) · 23000 > 2007 · 22007 1 % % 32007 − 23000 > 2007 · 22007

1 % & 4 P " 2 % , # & + "& & cm2 " 5 6 %& & + "& & % & &! " " ' & cm & 2P 1 2P 2P 2P 2P 2P 2 3 1 > 2 + 3 &! " " # " ' & 5 &! " & " ' / " % &5 & " / 2 ' 6 " % & & & " ' 6 *& & & ' 6 7 " # *& & " % ) % &2 & " " 2 " % ' 6 *& & & & & ( & " % " ' 6 a b 8


aabb abab abba baba baab bbaa 3 ¡ 6 18 ! "

# $ " ! " #

" ! " #

3 ¡ 4 ¡ 3 36 " %


20072005 −2007 = 2007·(20072004 −1) = 9·223·(20072004 −1), 20074 1 20072004 1 20072004 − 1 10 20072005 − 2007 !

9 10 " 90 ! # $ " " A B C % D & ' E " $ " " B '" " ABD ( ) ) ) EBD ) EDB " 30â—¦ )$ " BE " " )$ " % ") BD " 4 cm * + " '" " 16 cm2 *

+ ,) )- + ") ) " ) EBD . √ )$ " )$ ED . D

" 4 3 cm #" ). $ "

)$ AE " 8 − 30◦ √ 4 3 cm + ,) )- + . C ") ) " ) ABE

)$ " " " E ·· AB √ " . A B 8 2 − 3 cm * # " . ) " " √ √ " " '" √ √ 2 (8 2− 3) 3 cm2 , 32 3 − 48 cm2 * √

+ " " 4√ 2 2 32 3 − 48 cm + 3 · 16 cm " " 32 3 cm2 * / 01 2 ! 342 /! / 1 · 3 + 2 · 4 + 3 · 5 + · · · + 2004 · 2006 = (2 − 1) · (2 + 1) + (3 − 1) · (3 + 1) + (4 − 1) · (4 + 1) + · · · + (2005 − 1) · (2005 + 1) = 22 − 1 + 32 − 1 + 42 − 1 + · · · + 20052 − 1 = 12 + 22 + 32 + · · · + 20052 − 2005 3 $ " " 2005 3 * & ' F " $ " ' E " )$ AB 5 . ) AF E ABD ) ) ) " ' ) F AE ) '" 6 F E : BD = AF : AB 3 5 ) F BE ABC ) ) ) " ' ) F BE )


F E : AC = F B : AB FE +F B + FACE = AFAB

BD 1 1 F E · ( BD + AC ) = 1

F E 2 cm

C

D E · A

· F

· B

! " # $ " ! % " ! " # & " ! ' " ! " ( " !

" $ " ! ( ' ) % " ! $ ( " ! ( ' & $ # $ ( $ $ (& $ $ $ " ! % $ " ! * (1+2+3+4+5)·24·(10000+ 1000 + 100 + 10 + 1) 3999960 (


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