REG2007OU

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XXV

31.03.2007

2007 1

IV

1.

,

30. ?. ( XXXII-1, 6)

30 : 30=156

30=235. 30=156 :

(5+5) 30 = 235 :

(5+5) , .

2. . 160 , 40 .

. ,

20 .

?.(XXXII-2, 2493)

40 , 20 . ,

4020=20 .(7) , 160+20=180 .(6)

20 , 18040=140 .(7)

3. , . 75, 4 50 .

?

. 24:4 = 6, 6 .(6) 75 7

15 .(6)4 50 ,

7.15.+4.50.=12.5.(8)

4. 30 .

20

. 8 , 3 . ?

. (30-20):2=10:2=5 .(7),

30-5=25 .(5) , 8 , 3 ,

8 25 3 5 200 15 185 = = .(8)

5. .

.

.

.() AB , CD . ,

5

5 11 6 65 1 16 6 55 1 6

22 3 55 2 23 3 55 2 3

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2007 2

AB , CD ().(5)

CD

AB,

,

2 .(5)

AB

CD,

3 2 6 = , 6 2 8+ =

.(5), 4 6 3 8 24 = = .(5)

. a , b

. , 4 3 .a b= (5)

, ,b a> 2b a= + .(5)

, , 4 3 ( 2)a a= + ,

4 3 6a a= + , 6a= ,

8b= .(5) , 4 6 3 8 24 = =

.(5)

V

1. 300. ,

. ?

. ( XXXII-2, 2489)

300 b c+ + = , 2 b= , c b= + .

300 :2 150c= = , 150a b+ = .(7) , , 3 150b = , 150 :3 50b= = .(7) ,

2 2 50 100 b= = = . , 100, 50 150.(6)

2. ab, . c.

b..(XXXII-1, 11)

, ,

ab.()(5)

7 = , 180 + = , 8 180 = ,

180 :8 22,5 22 30' = = = .(10) ,

b90 90 22 30' 67 30' = = (90 90 22 30' 112 30' + = + = ).(5)3. 0, 1, 3, 4, 5.

, 4, 5.

. abcde ,

, , , , {0,1,3,4,5}a b c d e . abcde

, 0a .(2) 4 | abcde , 4 | de ,

A B

DC

a

b

c

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{04,40}de .(2) 5 |abcde/ , {0,5}e .(2), 04de = , .. 0d =

4e = .(2)1, 3, 5 .

{135,153,315,351,513,531}abc . , 13504, 15304, 31504, 35104, 51304, 53104.(2+2+2+2+2+2)

4..

6cm,

16cm. ?

. B, C,

D, E . M AC, N CD, P

DE Q EB. ,

6NP cm= 16Q cm= .()(5) ,

6NP cm= 2 6 12CD DE cm+ = = .(5)

16Q cm= , 16 12 4C EQ cm+ = = , M

AC Q EB, 4M QB cm+ = .(5) ,

( ) 4 16 20B AM QB MQ cm= + + = + = .(5)

5. O ABCD

POED, OMCE,

AKOP, KBMO ().

1) POED, OMCE, AKOP,

2, 4, 6,

D.

2) POED, OMCE, AKOP,

6, 10, 10,

D.

.1) OMCE

POED, . ,

AKOP POED, OK

OE. BMOK 2 3 6 =

POED, . 6 2 12 = .(5)ABCD

2 4 6 12 24+ + + = .(5)

2) OMCE4POED,

4:2=2 . , OK .

KBMO 2 2 2 2 8 + = POED,6 8 14+ = .(5) ABCD,

2,

ABCD 6 10 10 14

2

20+ + + = .(5)

A BC D E

M N P Q

C

O

D

K

P

E

BA

M

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VI

1. 56

.

10 , 78

.

?

. ( XXXII-2, 2501) x

. 5 76 8

10 x+ = , 7 58 6

10 ( ) x= (10)

.. 124

10 x= , 24 10 240= = . , 240 .(10)

2. a b ,

54a 63 1b 12.

.(XXXII-1, 2481) 54 63 1a b 12, 3 4.(2) 63 1b

b , 4, 54a

4.(3) 4, 54a 4

4a 4, 0a = , 4a = 8a = .(5)

0a = , 54 540a = 3, b

.(3) 4a= 8a= , 54 544a = , 54 548a =

3, 54 63 1a b 12 63 1b 3. ,

6 3 1 10b b+ + + = + 3, 2b= , 5b= 8b= .(5)

, 54 63 1a b 12 0a= {0,1,2,3,4,5,6,7,8,9}b

{4,8}a {2,5,8}b .(2)

3. 180 192 . ,

.

40 . ?. x. ,

,

.

180:m x= ,

192:d x= .(5),x180 192. 180 2 2 5 3 3=

192 2 2 2 2 2 2 3= ,(5) {1,2,4,6,12} .

40m d+ , 40m 40d . , 5x ( 5< ,

180 : 4 45m = 192 : 4 48d = ). , {6,12}x .(5) 6x= ,

180 : 6 30m = = , 192 :6 32d = = 62 40m d+ = > . 12= , 180 :12 15m = = ,

192 :12 16d = = 31 40m d+ = . , 15

16 .(5)

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4. ABC, B CB= ,

10C cm= . DABnAB

CBE. ABC40cm, AEC.

. ABC ( B CB= ), DAB, nABDCB

E.()(5) 10C cm= 40ABCL cm= ,

(40 10) : 2 15B CB cm= = = .(3)

ADEBDE(: D BD= D

AB, 90DE BDE = = , DE).

E BE= .(5), AEC ( ) 15 10 25AECL AE EC AC BE EC AC BC AC cm= + + = + + = + = + = .(7)

5. 20%

.

. ,

.()(5), 80 4

100 580% = = = .(3)

180 , 180 + + = 1 12 2

180 + + = .(2) 45

=

, 4 1 1 1 15 2 2 2 2

( ) ( ) 180 + + + + = ,

, 4 1 1 15 2 2 5

( ) 180 + + + = .

1 1 12 2 5

+ = . 1 12 2

180 + = .

15

180 = , 65

180= , 56

180 150= = . ,

4 45 5 150 120 = = = .(10)

VII

1. 2 2 2 2 2 2 2 28004 8003 8002 8001 4 3 2 1 + + + + .

. (XXXII-2, 2505)2 2 2 2 2 2 2 28004 8003 8002 8001 ... 4 3 2 1 (8004 8003)(8004 8003) + + + + = + +

(8002 8001)(8002 8001) ... (4 3)(4 3) (2 1)(2 1)+ + + + + + + = (10)

8004 8003 8002 8001 ... 4 3 2 1 (8004 1) (8003 2) ... (4003 4002)= + + + + + + + + = + + + + + + =

4002 8005 32036010= = (10)

n

E

D

C

B

A

A

B C

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2. , (

)

.

124?

. ( XXXII-1,

11) a .

. 2a

,

.

b , .. 2b

.(5) 124

2 2

124a b = .. ( )( ) 1 2 2 31a b a b + = .(5) 1a b = , 124a b+ = 2a b = , 62a b+ = 4a b = , 31a b+ = .(5)

, 32a= ,

30b = .. 32 32 1024 = .(5)

3. .

21mm ,23mm , 212mm . , 31mm 0,003

?. a, b, c .

1a b = , 3b c = 12a c = .(5)

, 21 3 12 ( ) ( ) ( ) ( ) ( )b bc ca abc abc V V V = = = = ,

V,

31 3 12 36 6V mm= = = .(10) ,

6 0,003 0,018 = .(5)

4. 1650m

.

, ,

11 . .

. 1v , 2v

. ( 1 1/60 ) 16501 1

1 260 60 1000v v+ = (7), ( 11 , 11/60 )

165011 111 260 60 1000

v v = .(7) :

1 2 99v v+ = , 1 2 9v v = , 1 54v = km/h, 2v =45 km/h.(6)

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5. BC. 1 1B

, . 2 2B

1 1B , B . 1 1BA B , 2 2B A B 1 1 2 2B A B .

. BC 1 , 1B , 2

2B .()(5)

B

, 1 1

,

, 1

1B

.(5) 2 2B

B

.(5) ,

1 1BA B , 2 2B A B 1 1 2 2B A B ,

.(5)

VIII

1. 3 4 .

, 69 . 3

4 .. (XXXII-1, 7)x

3, y 4 .

3 4 2( ) 69x y x y+ + + = , 5 6 69y+ = .(8)

. , 6y, x.

2 10 12 138x y+ = , 12 138 10y x= . ,

12y8 x12.

:x=3,y=9 x=9,y=4.(12)

2. 2a cm= ,

51 , , . 1cm .

. (XXXII-1, 5) .

2cm.

,

1cm. 1cm

1cm.(6)51 ,

. x

. , 51x x> , 2 51x > , 25> .(8)

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1cm 24, 25,

1cm.(6)

3. y kx n= + ,

, { 1,1}k n .

y kx n= + ,

.. :

1y x= , 1y x= + , 1y x= , 1y x= + .()(12)

2d = ,(4)

2 / 2 2P d= = ..(4)4..

. 75%

. ?

. x

y, y+2, y+4, y+6, y+8, y . ,

( 2) ( 4) ( 6) ( 8)

53

y y y y yx

+ + + + + + + += ,(5) 75

100 2

y xy

+= (5),

3 4x y= +

, 3 2 2y y x= +

, 4x=

, 8y=

. , 4.(10)

5. ABC ( C BC= )

B C D.

D

ABC.

. C BC a= = r

BC B E F

. AFO, FBO OBE ,

.F BF BE a r= = = (6) ,

ABC 2.AB a= (2)

B AF FB= + , 2( ) 2,a r a = (2)

2 22

r a= .(*) OBE DBC ,

a r ra CD

= , ara r

CD

= .(5),( 2 )a a rar

a r a r DA a CD a

= = = .

r(*), 2 2

2

( (2 2) ) 2 ( 2 1) 2 22

2

2 2a a a a

a a

DA a r

= = = = .(5)

REG2007OU

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