Geometrija - olimpijade

. ..

: .. , .. ,..

2007

51(07) 22.1

35

35 . .. / .

. . , .. , . . . .:, 2007. 152 .: .

ISBN 978-5-94057-280-0 . . .

(20052007) . .. .

, - . - .

22.1

ISBN 978-5-94057-280-0 , 2007.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 (2005) . . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (2006) . . . . . . . . . . . . . . . . . . . . . . . 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 (2007) . . . . . . . . . . . . . . . . . . . . . . . . 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

.. . ? . . . . . . . . . . . . . . . . . . . . . 93.. . ? . . . . . . . . . . . . . . 99.. . . . . . . . . . . . . . . . . . . . . . . . 120.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.. . .. ( ) . . 126.. . .. . . . . . . . 133.. . . . . . . . . . . . . . . . 137.. , .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

2005 . - , , - . . , - , . . , - . .

, 2005 . 2006 , III .

(2005)

1

1. (.) AC BD - P. AC BD C D, , Q. , AB PQ -. (8 )

. ( ). R DQ (. 1).

A

B

OP

D

C

Q R

. 1

1 . ..

6 (2005)

PDCQ ( PQ), - CDQ CPQ . CDQ = CAR, , , PQ AR ( ). BR , BD DQ, BAR = 90.

2. (.) , - , , .(89 )

. .

( ,. , . )

( ,. , 9)

( ,. , 27)

7

3. (.) K . - , K , . - (). (89 )

. OK ( O ) R/2 ( R ).

. , PQ ,M , O1 . - , OPO1Q , M OO1. MM1 OKO1

K

P

Q

O O1M

M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1M1

. 2

K

P

Q

O O1M

K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1K1

. 3

8 (2005)

KO1, . . . , OK R/2 (. 2).

, - .

. , - K , K R. O1 , , -, M OO1. , , O 1/2 (. 3).

4. (.) n n-, , ? (911 )

. . , - . , . -. , , f, 180 f. , , , . . , a, b, c, d . , ab + cd = ad + bc (a c) (b d) = 0. , , - . , , 60 (. 4).

60

120

. 4

9

. 108 , -

108

. 5

-, (. 5).

5. (.) - p1 p2. A B p1, C p2. BC ABC, -. , : ) ; ) ; ) .(810 )

. , - , (. 6).

A

B

C

C0

H

H0

p1 p2

A

B

C

C0

M

M0

p1 p2

A

B

C

C0O

O0

p1 p2

. 6

10 (2005)

, , , BC A. ,

A

B

C

A1

M

M0

p1 p2m

. 7

1 : 2, . , BC , , . - , - A BC, 2 : 1, , , . M0, AC0 ( - BC, A) 1 : 2, 2 : 1 , A AC0.

: A1 ( BC) . A1. , m1 . M p1 m. M, 2. , , p1 m, 2 : 1 (.7).

A

B

C

C0

K

O

O0

p1 p2

. 8

11

, AC BC , ( ) . , AK , AC0 ( - A B) A p2.

, , . , , ABC, p2 K , AK = BC. A K (. 8). , ( ).

6. (. ) AB ABC n ( B0 = A, B1, B2, . . . , Bn = B), AC n+ 1 ( C0 = A, C1, C2, . . . , Cn+1 = C). CiBiCi+1. ? (1011 )

. , - . B1, . . . , Bn -

h1

h2

h3

h4

A

B C

B1

B2

B3

C1

C2

C3

C4

S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1

2S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S12S1



S1

2S1

3S1

4S1

. 9

12 (2005)

AC. CiBiCi+1 , , , hi = ih1. , -: Si = iS1. ( . 9 n = 4.)

C1, . . . , Cn AC , . , ( , h1 ). - -, -.

, - ( ) , , BiCi ( , ). SBi1BiCi = SCi1BiCi( BiCi , , ) SCi1BiCi = SBiCiCi+1 ( Bi , Ci1Ci = CiCi+1 ).

7. (.) 1 2 O. A ABC , BC . BOC? (89 )

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

A1

B1 C1K

I1O

A2

C2B2K

. 10

13

. 60, 120.

. , B C ,

A

B C

M

O1

K

I1

RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR




. 11

(. 10). ,

( )

B1OC1 =12B1I1C1 = 60

,

-. B2OC2 = 120.

- . I1 , AB AC ABC B C , K BC. K AI1 3 : 1, I1K (. 11).

(. 12).

O

A

B C

P

Q

M

K

I1

O1

. 12

14 (2005)

AO P Q - . M, O1 K AI1 , R/2, R , AB AC B C. , I1O = R. , P O AQ , , ,, MP KQ . PKQ , MPK .

, - , ,

O1P = O1M = O1K =R

2.

, O1P AI1O , , I1O.

. ( , . , 9) -, , , O A - BC (. 13).

O

A

B CK

G

O1

. 13

15

K BC, G - ABC. OK O1. BK = KC, OK = KO1, - BOCO1 . , , G ( ) AOO1, AK , AG : GK = 2 : 1. , OG , AGO O1GO. ,GA = GO1 = GB = GC, . . A, B, C, O1 , . .

BO1C = 180 BAC = 120.

, 60.

. ( , . , ) , , - . AB l. X -, AX/BX = l, . P Q , AB l ( ), -, PQ . (. 14).

O

A

B CK

G

. 14

16 (2005)

,

AG

KG=

AB

KB=

AC

KC=

AO

KO= 2,

, B, G, O, C AK l = 2. , BGC = BOC (180 BOC).

8. (.) ABCD . , -

P Q

RS

A

B

C

D

. 15

. , - ? ( ABCD, ). (89 )

. . . A C , B D . AC BD AC = BD (. 15).

, - . , , A , BD,

P1 Q1

R1S1

P2

Q2

R2

S2

A

B

C

D

C1

C2

. 16

17

C1 C2(. 16).

, AC1 = BD AC2 = BD, . . AC1 = AC2 C1, C2 . , , C. , AC, , , ABCD -. , , .

9. (.) O ABC. P - . M - ., M PO. (9 )

. , 2PM =

PO. ,

G ABC, P

3PG =

PA+

PB +

PC.

. ABC

P. PA,

PB,

PC, na (P),nb (P) nc (P), P, -

A

B C

P

. 17

18 (2005)

, P .

2(na (P) +nb (P) +nc (P)) =PA+

PB +

PC.

, - , P (. 17).

P, . ( , P .) , P , , - . , . , , P ABC.

10. (.) , .(89 )

. AB 6= AC. BC , ACB = ACB (. 18).

, ABC ABC , BC BC. BC M, ABC ABAC. A1 AM BC AB1A1C1. A1C1, B1A1 B1C1 .

. , . ,

M

A

A

A1B

B

B1

C

C

C1

. 18

19

, . ( BC ) PQ c AB AC, BC. , APQ = ABC AQP = ACB (.19). ( BC -) RT c AB AC , ART = ACB ATR = ABC. ( , , , , -.) , , . , - .

. , (. 19).

A

B CA0A1

R

T

PQ

OO1

. 19

(. 20).


A

A1B

B1

C

C1

. 20

20 (2005)

, AB 6= AC. AA1 ,A1C1 A1B1 AC AB . - A1C1AB1 , - , . . C1B1 , , , C1B1 . , A1B1C1, AB1C1, C1BA1 B1A1C ABC, . ( A1 BC. , BA1/CA1 = AB2/AC2 .)

11. (.) n ai bi,i = 1, . . . , n. n a1, . . . , an, b1 . . . , bn ? (810 )

. n = 5. , ( , ) , , . -, , . , , -. - ( - ), ( , ).

, , . : - ( - ), . , (. 21).

A B

CD

P

12

3

. 21

21

1 12

2 11

3 9

5 13

7 10

. 22

( , - , -, ). , P. C, , D , , .

5 - (. 22).

12. (.) a, b, c d l . (810 )

. ABCD , P, Q, R, S,X , Y AB, BC, CD, DA, AC, BD . QX , SY ABC ABD,

QX = YS =a

2.

,QY = XS =

c

2.

, X Y XYQ XYS, Q S. P R.

22 (2005)

A

B

C

D

X

Y

Q

S

PR

a

b

c

d

llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

a/2

c/2

. 23

P, Q, R, S , QX , PX , QY ,PY , (. 23).

13. (.) ABC l1, l2. D AB , - l1, AC E , , l2, - BC F . D, EF . (9 )

. P - AC E BC F . D AB, DEPF , , , . , CP, CEF , (. 24). , , . . C, Q. EF C, - CEF . , , , , , CQ .

, , D. A , l2, U - BC. B , l1, V AC. Q , ACU BCV , E -

23

A

B C

DE

F

P

OD

. 24

AC CQ. , E l1, AB .

14. (.) P - ABC. A1, B1 C1 AP,BP CP BC, CA AB. - AB1C1, A1BC1, A1B1C, S1, S2, S3. ,

S1S2 6 S 6S2S3,

S A1B1C1. (1011 )

. . A1B1C1 -

P. , ABC - - ABC , P , - P (, , . 25).

, BC A1,

BA1A1C

=

BA1A1C

,

BC. - A0,

BA0C=

p

2

24 (2005)

A

B CA1

B1C

1

H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H H H H H H H H H HH H H H H H H H H HH H

A

B CA1

B1C1

P

. 25

( , BC

, . 26)., A -

BB1 AC. A A0,

B CA1

A

A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0B

1

A

B1

. 26

CB1 /B

1A , A0 -

, . , - A ,

CB1B1 A

=

CB1B1A

.

- . -, ABC ABC -, - . - , ,

, - . , . , . , AB1C1, A1BC1, A1B1C S1, S2, S3. - cosA, cosB, cosC -, , ,

25

. ,

C 6p

36 A.

, S1S2 6 S.

S2

cos2 A cos2 B

(1 cos2 A cos2 B cos2 C)2 6 1.

, ,

1 cos2 A cos2 B cos2 C = 2 cosA cosB cosC, 1/4 6 cos2 C, . . C 6 p/3. ,

S2S3 > S.

. , AB1C1, A1BC1, A1B1C S1,S2, S3 (. 27).

P ( ABC) - (p, q, r), . . p+ q + r = 1. - P , p, q, r . S1/S. A2 - B1C1 AA1. AB1C1 A1B1C1 , , ,

S1S=

AA2A1A2

.

A

B CA1

B1

C1A2

P

. 27

26 (2005)

, , A2 (2p : q : r) ( 2pA (q+r)A1 AA1, (p+q)C1 (p + r)B1 B1C1),

AA2A1A2

=q + r

2p=

1 p2p

.

,

S2S=

1 q2q

S3S=

1 r2r

.

S1 6 S2 6 S3, , p > q > r. p + q + r = 1 p > 1/3 > r. , S1S2 6 S. ,

(1 p) (1 q) 6 4pq,. . r 6 3pq.

pq

r>

13 qr>

13.

, S2S3 > S (

r 6 1/3).. , , -

, . ,

a =BA1CA1

, b =CB1AB1

, g =AC1BC1

( ) .

. ( , . --, 5) . P ABC. A1, B1 C1 AP, BP CP BC, CA, AB, AB1C1, A1BC1, A1B1C A1B1C1 S1, S2,S3 S .

S3 + (S1 + S2 + S3)S2 4S1S2S3 = 0.

( , , , - .)

(x) = x3 + (S1 + S2 + S3)x2 4S1S2S3.

27

(S) = 0. , , (x) - (0,) ( ). ,

(S1S2) 6 0 6 (

S2S3).

(S1S2) = S1S2 (

S1S2 + S1 + S2 3S3),

S1S2 + S1 + S2 3S3 6 32 (S1 + S2) 3S3 6 0

( ). -.

15. (.) ., , . (11 )

. , 1/

2 p/4.

, , . , - , (1/2, 1/2). - . 28 , .

. 28

28 (2005)

, (x, y) -

(x, y) =(x y2

,x + y

2

).

, x y , x, y

x2 + y2 =x2 + y2

2=

R2

2.

, x y , (1/2, 1/2) (x, y) (

x 12

)2+

(y 1

2

)2=

R2

2.

16. (., .) 4 : -, ( ) (- ). . , , - . . (89 )

. , , , arccos(1/4). ABC , A1, B1, C1 BC,CA AB . ABC A1B1C1 M ( 1/2), O ABC - A1B1C1, M OH (H ABC) HM = 2MO (, , ABC).

I ( ) , - ABC. , , , A B .

OBA = HBC =p

2 C,

BI HBO. , I OH , OI = 2IH ( ).

29

, BO = 2BH . -, , AO = 2AH . ,

AH = BH =R

2,

R ABC . -, , , , AH = 2OA1 ( ). , OA1 = R cosA.

AH = 2R cosA cosA =14.

, cosB = 1/4.

17. (.) ABC - I P, Q, R BC, CA AB. K , -, B C, ( ) . (11 )

. , - , , . - , , . -, . , ! -, AB = AC, (K , P), , AB 6= AC.

.1. BC.2. QR T

BC.3. Pd , P.4. PdT K -

. K (. 29).

. , T BC , P ( ). . KP BKC ( :

30 (2005)

A

B CP

Q

R

T

PdK

I

(1)

(2)

(3)

(4)

. 29

B C; , - K , BC P; KP BC) 1.

BP

CP=

KB

KC= l 6= 1.

K (. 7, ) BC l, PT , . . TKP = 90 , , PKPd = 90. .

18. (.) l1, l2, l3, - , O . O Oi , O li .

) , M , O1O2 M1M2, O2O3 M2M3, O3O1 M3M1, - . (1011 )

) ? (1011 )

. . , -

, . (, - ABC , -

1.: , , 6, .

31

A

B C

A1

B1 C1

H

O

O1

O2O3

. 30

, . . . , .)

ABC , li , H . O2O3, O3O1, O1O2 AH , BH , CH ( A1, B1, C1), , ABC. -, O1O2O3 ABC , O 2. , - O1O2O3 ABC. , , C O1O2, , O3 AB, . . ABC, H ABC A1B1C1 (. 30).

, , M , D M1M2.

DC1 =

DO1 +

DO2

2

, M1O1

M2O2

C 2C,DC1 , C.

32 (2005)

A

B C

D

O

M

O1

O2

M1

M2

C1

. 31

,M1O1

M2O2

MO

CB CA , DC1

MO

C ( , C1 A1B1C1, (. 31).

, , , A1, B1, C1 M1M2M3, - A1B1C1 , A1, B1, C1 OM. .

. , -, , , , , , . ( ).

A1B1C1 , . . .

33

A

B C

M2

M3

M

M1

P

Q

. 32

. ( , . --, 5) , ABC , - li , H A, B, C , BC, CA, AB . . 1 2 H, . , 1 2 1 2, H 1 2. , M1 ( M3M2) M ABC ABC ( , 1/| cosA|, A A . 10). ,

AM =AM1| cosA|

( -) M1 AB AC M (. . AM1 A AM, . 32).

P, Q M AB AC. M1Q - M2MM3, AB. M1P AC, M

1 APQ, ,

AM1 = 2r cosA,

34 (2005)

r , APQ. ,

r =AM12

.

- . ,

H

A

B CA

BC

O1

O2 O3

. 33

M2 M - ABC ABC, M3 M ABC ABC., M - O ABC , , , - O1, O

2, O

3 -, H ABC -. ( , H O, - - , , - 16 , , -

O1 AH . , AO = R AH = 2R cosA,, AO1 = AH/2 . .). - , O1M

1, O

2M

2 O

3M

3 - ABC, ABC ABC (. 33).

, ( - , , , ), O1A

, O2B O3C

, - ABC. , , O1M

1 O

1A, O2M

2 O

2B, O3M

3 O

3C .

, , O1M

1, O

2M

2 O

3M

3 - , .

19. (.) , . , , . . - ( A)

35

Z

X Y

T

L1 L2

OxOy

. 34

( A) , AA , . .

) , - , . (811 )

) , , - , , , . (811 )

.a) -

. : , L1, L2.

2L1L2.

X Y

L1L2, XY = 2

L1L2. , , ,

, XY , XY/2, Z, , .

X Y - 2. , -

36 (2005)

, R Ox , Oy , (. 34)

OxOy < XY + 2(XZ + YZ) < 4R.

) L0, L1. , - .

2ZL, T

t0t1 Z , , . T . , ,

kT ( T ,

, )., X Y , XY , ki

Ti ,

ki Z, Ti t0t1, (. 35).

XY v1 + v2 ,

.

, v1 = l(t1 t0), v2 =XY v1 . v1 l

X

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Z

L0

t0

L1

t1

t1/2 T1T2T3

T4

v1

v2

. 35

37

l v2 v1 , - l v2 . , m1 m2 T1, T2, T3, T4, , v1 = m1 (

T2

T1)

v2 = m2 (T3

T4). .

20. (.) I , ABCD, A, B, C, D , IBCD,ICDA, IDBA, IABC . , , ABCD, , ABCD.(811 )

. R, r ABCD, O ABCD, L ABC, H I ABC. , O D ABC, L, OD IH . , DA = DI ( , IABC), OA = R,IH = r.

ADO ODI :

R2 = DA2 +DO2 2DA DO cosADOOI2 = DI2 +DO2 2DI DO cosIDO.

, :

R2 OI2 = 2DO (DI cosIDO DA cosADO).

,

DO =R2 OI2

2r.

, A, B, C O . , ABCD ABCD (. . ), DO = r , ABCD. , r > R. DOI . O,I R, r, . D , . , .

38 (2005)

, D DX1 DY1 -, , - DX1Y1. , . , , ( r) DX Y , - D. , , - ( ):

OI = R2 2Rr.

,

r =R2 OI2

2R.

, r > r. .

21. (.) , , 900 . 2 ) , ) , 300 /? (1011 )

.) -

,

A

B

C

D

O

X

O

A B

C DD

D

X

. 36

39

A B

CDD

D

P Q

O

. 37

. O ABC, X ABD O X - AC. - , AD. AB, X (. 36).

, , 600 - (. 37).

cosPOC =OC

OP=

900

6003=

32, , POQ = p/3.

:

S =12 p3 6002 S = 600

2

2

(p

332

).

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C DD

DC C


P

Q

. 38

,

p 6002 32 6002

(p

332

)=

= 180000p + 2700003.

) - , , - - (. 38).

, -, .

POA= arccosOA

OP= arccos

34, PQ=2PO sinPOA=300

7,

S=2180000 arccos 34675007,

S=p 6002720000 arccos 34+135000

7=720000 arcsin

34+135000

7.

40 (2005)

22. (.) ( ). , .(1011 )

. ABCD , A, B, C, D BCD, CDA, DAB, ABC. , -, , - . , , ABC ABC . .

, P Q AC AB. 2 : 1, , - , BC PQ. PQ BC , ,BC BC. AC AC, . , A, B, C, D ABCD, - ABCD. - ABC , , . ABC .

23. (. .) - . (1011 )

. , , - (. 39).

. 39

41

24. (.) , f, f < 2p/3. , , f. (1011 )

. . ABC .

, f., BC, CA, AB X , Y , Z -, AZ = AY , BZ = BX , CX = CY . AC , AB . - Z AB, BC X , BX = BZ (X , AB 6 BC), Y , B AC, AY = AZ, CY = CX . Z = B AY = AB, CY = CB. , AYC = B < f Y , AC (. 40).

Z = A Y , AC > BC. , Z Y AC (. 41).

, ABC, ABZ, BCX , ACY , . . A,

A

B C

Z

Y

X

. 40

42 (2005)

A

B C

Z

Y

X

. 41

A

B C

K

L

M

T

120 120

A

B

C

. 42

43

B, C ABC . , A + B + C > 3p 3f > 3p 2p = p . ( : , ).

. ( , ., 192) AB, BC CA , f 6 . BC wA, AC wB AB wC . K , L, M (. 42). , -, (, , , 2p/3).

, M, L, K , wA,wB , wC . , , - BC f , wA BC. FA. FB , FC. FA FB , C K , K , FC, C . , FA FB FC. .

9

1. (..) ABCD , O . , BAO = DAC, .

.

ABO =p AOB

2=

p

2 ADB,

DAC + ADB =p

2,

(. 43).

2. (..) , - - .

44 (2005)

O

A

B

C

D

. 43

. - . ABC (C ), AB A, B ,

AB = BC = CA = AB,

ABC, ABC ABC (. 44).

ABA B

C

. 44

, - : , , (. 45).

, ABC (AC = BC) .

AXB - AX = BX ,

45

. 45

. - . , , - C D , AD = AC. , , , .

3. (..) O A, B . , AB, .

. K -. , K , OK . - AB , OKA, OKB , . , , OA, OB (. 46).

A

B

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK

. 46

46 (2005)

4. (. .) P - ABCD, M , , O - , H , APD BCP, APB CPD. -, M OH .

. O1 AC, O2 BD. , M O1O2 (, M 1A, 1B, 1C, 1D, . . , A, B, C, D. 1A, 1C 1B, 1D, 2O1, 2O2).

, H1H3H2H4, -, , -, -. H - .

, HO1 OO2, , , BD. , H , O1. AH4 K . AH3H4, K

A

B

C

D

P

MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMO

O1

O2

HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

H3

H4

H1

H2

K

. 47

47

AH4. , AH4C, O1.

, , HO2 OO1, . . HO1OO2 , M (. 47).

, O, M, H OM = MH .

5. (..) , , . , 135.

. , . - 60, , , . , 60. 60, - , . . .

A , B C ,AM , AH , M BH . AMH < 30. ABM BAM. . ABM < 15.

B 60 , BC. ACB < 30. ,

BAC > 180 15 30 = 135.

10

1. (..) - . , , . , .

. D ABCD

ABCD. BCA < CAD BAC ACB, AXB AXC

49

BXC. ,

AXB = AXC = BXC = 120.

AX = BX = CX ABC . ,

, , . , , . . - , . - , , , . , .

3. (.. ) O AB CD. AB CD

O

A B

C D

X

Y

P

Q

. 50

P. , OP AB CD.

. X , Y - AB CD, Q OP.

XQ2 =2OX2 + 2XP2 OP2

4=

=2OX2 + 2XA2 OP2

4=

=2OA2 OP2

4= YQ2.

, Q - X Y , , AB CD (. 50).

4. ( , ) A1B1 A2B2,

A2B2A1B1

= k < 1.

A1A2 A3, A2 A4 ,

A3A2A3A1

=A4A2A4A1

= k.

50 (2005)

, B1B2 B3, B2 B4 ,

B3B2B3B1

=B4B2B4B1

= k.

A3B3 A4B4.

. O - , A1 A2 B1 B2. -

O

A1

A2

A3

A4

B1

B2

B3

B4

. 51

OA1B1 OA2B2 , A1OB1 = B2OA2 A1OA2 B1OB2 .

OA2OA1

=OB2OB1

= k,

- A1A2 B1B2 A3 B3, - A4 B4 (. 51). -, .

O, - A3A4 B3B4 -. A3A4 ,

A2 A1 k, , A1 A2 B1 B2. , .

. A1B1 =

u , A2B2 = v , , v 2 = k2u 2.

A3B3 =

A3A1 +

A1B1 +

B1B3 =

11+ k

A2A1 +

u + 11+ k

B1B2; ()

,

A3B3 =

A3A2 +

A2B2 +

B2B3 =

k

1+ k

A1A2 +

v + k1+ k

B2B1. ()

() k1+ k

, () 11+ k

,

A3B3 =k

1+ ku + 1

1+ kv .

51

A4B4 =1

1 kv k

1 ku .

(A3B3,

A4B4

)=

(ku +v , v ku )(1+ k) (1 k) =

v 2 k2u 21 k2 = 0,

. . .

. (. ) A1A2B2X A1Y A1XB1.

A1

A2

A3

B1

B2

B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3B3

X

Y

. 52

B1Y

XY=

A1B1A1X

= k,

B3Y B2X B3Y = kB2X = A1A3.

, A1A3B3Y -, . . A3B3 A1Y (. 52).

, A4B4 - XA1B1, , A3B3 A4B4.

5. (..) 1 - X , Y , 1. C CA, CB . CB - A. AA.

. O , - C, O . OO =

3, AB,

B CA c , C. ,

AOA = AOC +12COB = 2ABC + CAB =

= CBA +12CAB,

OAO = OAB + BAO =p

2 COA + p

2 BCA =

= p BCA 12CAB = CBA +

12CAB.

52 (2005)

A

A

B

B

O

O

C

C

. 53

OA = OA, AOAO , AA == OO =

3 (. 53).

6. (..) H ABC, X . XH AH , BH , CH A1, B1, C1, AX , BX , CX A2, B2, C2. , A1A2, B1B2, C1C2 .

. , A1B2C1A2B1C2. XH = d .

A1B2 = d sinA1HB2 = d sinXBC,

HA1 BC, HB2 BX . -,

A1B2 C1A2 B1C2A2B1 C2A1 B2C1

=sinXBC sinXCA sinXABsinXAC sinXCB sinXBA = 1,

(. 54).

53

A

B

C

A1

B1

C1 A2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2B2

C2

H

X

. 54

. , A1B1C1 - ABC, , - X .

11

1. (.. ) A1, B1, C1 ABC. , A1,B1, C1, B1C1, C1A1, A1B1 A2, B2, C2 -

A B

C

A1B1

C1

A2

B2

Z

. 55

. , AA2,BB2, CC2 , - ABC .

. Z - AA2 BB2. B B1 A1C1,

ABZ = C1BB2 = B2B1C1.

BAZ = A2A1C1. A1A2 B1B2 -, A2A1C1 = B1B2C1, ,AZB = ACB A, B, C, Z - , (. 55).

54 (2005)

2. (..) ABCD. BC AD O, B OC, A OD. I OAB , J OCD , - CD . , IJ BC AD, ( ) X Y . , XY ABCD , BC AD XY .

. , , - , , , X Y

AD BC , OX + OY = l , l , .

M IJ . ,

OX +OY = l.

MXX MYY , , MXY MX Y . , X Y , MX , . . X X (. 56).

A

B

C

DO

I

JM

X X

YY

. 56

3. (.. ) ABCD K , ABCD - . , K ABCD.

55

. U ABCD A C, X , Y U AB BC.

UX

UY=

sinUAXsinUCY

=sinBCAsinBAC

=AB

BC,

. . K UB. , K UD,, , K = U . -, , AV CV , V B D, K = V , . , B, D, U . AB/AD = AU/UD = CU/UD = BC/CD A, C, V . , K AC BD.

. , AB CD , , AB CD. ABCD, LAB LCD, L -, , . . L . , L , , K .

4. (..) ABC A = a, BC = a. AB AC M P. , MP BC.

. B C MP (. 57), . .

12(BM sinAMP + CP sinAPM) =

12(BM+ CP) cos

a

2=

a

2cos

a

2.

, a sina

2.

A

B C

PM

. 57

56 (2005)

. I -, X , Y BI , CI MP.

MXB = AMP MBX = C2.

A

B C

PM

I

XY

. 58

, BXM BCI -, . .

BX

BC=

BM

BI= cos

B

2.

, BXC (. 58)., BYC . -

,

XY = BC sinXCY = a sina

2.

5. (.. ) K . , - M, -: ,

K , A B, MK AMB.

. , - K A B, K , AK /BK = AK/BK . K - l , , KK

A

B

K

M

K

l

. 59

57

M K l . AM/BM = AK/BK , . . M (. 59).

. (.) O . - KXOY , . M , K XY . , M .

, K , OX OY A B. ,

MX = KX , MY = KY , MXY = KXY = OYX , MOYX MXO = MYO. ,

MXA = 180 MXO = 180 MYO = BYM., AXK KYB , - , KX/XA = BY/YK .

MX

XA=

KX

XA=

BY

YK=

BY

YM.

MXA BYM , MXA BYM (. 60).

O

A

B

X

Y

K

M

. 60

, ,

MA

BM=

MX

BY=

KX

BY=

AK

KB,

, MK AMB.

58 (2005)

6. (..) , ABCD, A, B, C, D. AA BB , . , CC DD .

. AA BB , AB AB . - (, ) P. P -

A

AB

B

. 61

- CD, CDP . , - - . - AB , AB . AA BB , AB ( . 61 - ABAB).

, ABC ABD 60, - , C D, 120, , C, D ABC, ABD, ( . 62 , AB).

AB

C

D

. 62

(2006)

1. (.) , - 46, F . ? (8 )

. 90.

. l1, l2 F . - l1, l2 l1, , l2 l1 (.63). , - F , l1 46, 2 46, . . . , n 46, . . . 46n n < 90 180, n = 1, . . . , 90, . . F 90 .

l1

l2

. 63

60 (2006)

, 90- 90 .

2. (.) A, B . , AB . (89 )

. ( 1543 .) l , - , A , A l . B A - , , AB . l AAB, -, AB .

3. () , , . , , , . , ?(89 )

. P. P k A, B, C, -, O ABC. P 1/k O .

4. (., .)) ABCD DEFG, E

CD, F , G ABCD. AE BF. (8 )

) OKLMN OPRST , - P ON , R, S, T OKLMN . KP MS. (911 )

.) H

(. 64).

AHD = 45, DHF = 90, EHF = 135,

61

A

B C

D

E F

G

H

. 64

K

L

M

N

O

P

R

S

T

. 65

A, E , H . , B, H , F - . , BHA = 45.

) . 72. . ) (. 65).

5. (.)) 10 10 1 118.

(8 )) 1010 1 (100+9

3)

( 1 115,58). (911 ) , .

62 (2006)

.) A, B

10 , C, D 12. , . 66, , - CD, , AB. 9 , .

A

B C

D

. 66

) A, B 10 , C, D 10 +

3.

, . 67, , CD, , AB. 9 , .

A

B C

D

. 67

6. (.)) AB C ,

R. , C .(89 )

63

) AB D , - r. A B , - . (910 )

. .

. , , AB, X , C. XC AXB.

. O , O , L AB, X (. 68). O OX , OC OL, OCX OLX . , C XL XC AXB .

A B

X

C

L

O

O

. 68

.) X .

, AX/XB = AC/CB. , - , AB ( A B). - , , C AB. . , , AB .

) . ) - X . -

64 (2006)

A, B. .

7. (.) ABCD E , F , ABE BCF . - ABE , , EF , BFD . (810 )

. BFD , F , . . BFC = 135 = AEB ( AEB, , ).

ABE = CBF 1, EBF = 90 BE

EF=

12=

BE

AB.

ABE , ,

sinEAB =BE sinAEB

AB=

12.

, EAB = 30, EBA = 15 (. 69).

A

B C

D

E

F

. 69

8. (.) AB , . r1 r2, r1 > r2. AB. (89 )

. AB , , r1 = r2,

1 , ABE = BCF .

65

A

B

. 70

. - , AB (. 70), . , AB - (. 71).

r1, r2 - - ABC. ,

r1 =AB + BC + CA

2, r2 =

BC + CA AB2

AB = r1 r2.

A

B

C

. 71

9. (. -) L(a) - , a p 2a. , a+ b+ g = 2p, L(a), L(b) L(g) 1.(810 )

. A, B, C , a, b, g. AB , (a+b)/2 = pg/2, L(g) , (g+ p 2g)/2 = p/2 g/2. , L(g) ABC. , L(a), L(b) ABC, , .

1 .

66 (2006)

10. (.) n n- - n 2 (,

. 72

. 73

, ) ?(811 )

. n - , ( n ).

. - ABC, (.72). - AB - , -, AB. -, AB - , . - -, , , - AB, . (AB , 20 .) BC AC. ABC ,

n = 2k + 2k + 2l = 2k+1 + 2l .

, n = 2k+2l , k > 0. A n-, B C 2k1 . AB = AC, , ABC (. 73).

11. (.) ABC O ; A, B, C , A, B, C ; A1, B1, C1 OA

BC, OB AC, OC AB. , AA1, BB1, CC1 . (910 )

. Oa, Ob , Oc , O - BC, CA, AB. , COc , OC AB , , AOa, BOb COc .

OaObOc - ABC O 2,

67

ABC, , , . , 9 .

12. (.) ABC A , A. , A , AB = AC. (910 )

. , . H , L, M , , P BC , A(. 74).

A

B C

P

M

H L

. 74

A , PM > AH . L AP, , HL < LM, , AL AHM, , , AH AM .

13. (.) a b, A B. X a, Y b , AX BY . AY XB. (910 )

. A , b - a U . , B , - a b V . X , Y AUX YVB. , , . .

68 (2006)

AY , BX UV . , UV .

14. (.) - P. - ABP, AB . (911 )

. O , C , A, B,C ABP, A, B, P; P C OP (. 75).

CCO = CPO = 90,

O, C, C, P , CP PC = OP PP. -, CP PC = BP PA. A ,,

BP AP = |R2 OP2|. , OP PP, , P, AB, . . , P OP.

P

P

A

A

B

B

C

CO

. 75

15. (.) ABC , BC, CA, AB A1, B1, C1 . B1C1 BC P, M PA1. , , M - . (911 )

69

. AB < AC. AA1, BB1 CC1 , ,

PB

PC=

A1B

AC.

,

MB =PB A1B

2, MC =

PC + A1C

2,

MA1 =PB + A1B

2=

PC A1C2

.

,MB

MA1=

MA1MC

=A1B

A1C,

.

16. (.) ABC . , . , - ? (911 )

. .

. . - ABC, , A > 60. BC ABC,

A

B C

A

B

C

. 76

70 (2006)

ABC = ABC = 60, BA , , AB BAC (. 76). -, AC , BAC = BAC = 60.

17. (.) , A B, A1B1 A2B2. AA1 BB2 X , AA2 BB1 Y . , XY A1B1. (911 )

. , A, B,X , Y , . . XAY = XBY .

XAY = BAA2 BAX = BAA2 BB1A1,XBY = B2BA AA1B1.

A1B1 A2B2 ,

ABB1 + A1B1B = BAA2 + B2A2A,

, , (. 77).

A

BA1

B1 A2

B2

X

Y

. 77

18. (.) H ABC , BC - X , AC Y . AZ, BZ HX HY . , X , Y , Z . (911 )

. , - . 78. U HX BZ, V HY AZ.

71

A B

C

H

XY Z

UV

. 78

HU

UX=

YV

HV

HU

YV=

HV

UX.

AYV BUH AYV BUH , . ,

HU

YV=

BU

AV. ,

HV

UX=

BU

AV.

-.

19. (. ) T , . T1. , T1 , - T .(1011 )

. T1 - T0, T , , .

O

O0O1

H

M

I0I

. 79

T0 T1, . . I0 T1, O0 - - , - T1, , , I0O1, O1 - T1. , O0 OH , O, H - - T , T M HO 2 : 1 (. 79). I0 1/3 I - T M, O0 3 M H . O1, O1 IH .

72 (2006)

20. (.) A, B, C, D. A1, B1, C1,D1 BCD, CDA, DAB, ABC. A2, B2,C2, D2 B1C1D1, C1D1A1, D1A1B1, A1B1C1 . . , , , . (1011 )

. , , ABC, BCD, CDA DAB, . X 9 - ACD BCD, AB; Y , Z, U AC, BC, CD.

YXZ = YXU + XUZ = DCA+ BDC = BCD,

. . X 9 ABC. , X 9 ABD. , - 9 CDA ACB1 , X 9 ABB1 CBB1. 9 ABA1 BCC1, , , 9 A1B1B, BB1C1 A1B1C1, .

, . U , V , W .

UVW , 9 .

, A, B,C, D, , .

21. (. ) AB, BC, CA ABC C, A, B. , :

SABCS2ABC > 4SABCSBCASCAB ,

, AA,BB, CC . (1011 )

. P1 = AB BC CA, P2 = BA AC CB. , SABC = (P1+P2)/4R, R - ABC, , ,

SABCSBCASCAB

SABC (SABC)2=

P1P2(P1 + P2)2

614,

P1 = P2, - AA, BB CC .

73

22. (.) , A, B P. X , Y AX BP. , PXY . (1011 )

. Q X - ABX PXY .

ABQ = AXQ = YXQ = YPQ = BPQ.

,BQP = p (BPQ + QBP) = p ABP

, , X . PXY Q PQ.

23. (.) ABCD , G (. . , , ).

) ABCD O. H G, ., H , G, O HG : GO = 2 : 1.(910 )

) ABCD I . - N , - , - . ( ). , N , G, I , NG : GI = 2 : 1. (1011 )

.) Ma Ha -

BCD. - . - O. ,, MaMbMcMd HaHbHcHd O 3. - .

) M1 -. G IM1 2 : 1.

74 (2006)

, M1 , , , G , IAB, IBC, ICD, IDA , - . , I 2/3.

a, b, c, d A, B, C, D. , , A, B, C, D a,b, c, d , N , , 2a+b+d , 2b+a+ c, 2c+b+d , 2d+ c+a, M1. , I b + d ,a+ c, b + d , a+ c.

I SIAB SIBC + SICD SIDA = 0. U V -. , ( ). X , Y BC AD. XY - L . LXB LYD, , BL/DL = b/d . ,AL/CL = a/c.

SUBCSUAD

=BL

DL,

SVBCSVAD

=CL

AL,

SIBCSIAD

=b + c

a+ d

, I AC (a + c)/(b + d), .

24. ()) P -

, - A B. P AB.(910 )

) P , A, B,C. P ABC.(1011 )

.) P1 , P AB,

P2 , P AB. - ABP1 ABP2 AB, , OP1 = OP2. APBP2

75

, OA2 + OB2 = OP2 + OP22 , . . OP2 PA, PB. , P1, P2 O, P AB OP.

) PABC PACBCBPA. . ) , OP2 = 3R2 2OP2,. . P O. M ABC PP 1 : 2,M , , OP 2 : 1. , O ABC O ABC , P H . M OHMH = 2MO, MK = KH , . . K

3R2 2OP2/3.

25. (.) ABCD BC, CD DA a, b. AB/CD. (11 )

. - A B, C D. ,

CBD = CBA = DAC = DAB,

ADB = CDB = DCA = BCA,

.

AB

BC=

BC

BD=

BD

CD=

sinBACsinBAD

=sin asin b

.

,AB

CD=

(sin asin b

)3.

26. (.) - (, , ). . , . (11 )

. - , . , - , . , , .

76 (2006)

8

1. (.) .

. , : ,

A

B

. 80

. 81

, , , , . . - A, B . AB . , - (. 80).

, , . 81, - . , .

2. (.) - n n-, , -, ..., 2006-?

. n = 3.

. . 82 , n > 3 n- (n+1)-. , - , ,

. 82

77

1002 , , , ..., - 2005- 2006-.

3. (. ) ABCD. A C D. l D X , Y . , BX = BY .

. , , , . 83. AX = AD = BC CY = CD = AB. ,

BCY = C DCY = C (p 2CDY) == 2CDY D = CDY ADX ,

BAX = DAX A = p 2ADX A == D 2ADX = CDY ADX.

, ABX CYB , -. X , Y -.

A

BC

D

X

Y

. 83

4. (.) A B. P A B , X , Y PA, PB . , , P AB, XY .

78 (2006)

. , P - (. 84). Q , - P AB, .

QPX =

QX + AP

2, QPY =

QY +BP

2.

AP BP

2= PBA PAB = QPX QPY ,

, QX QY . .

A

B

P Q

X

Y

. 84

5. (., .) , - ,

A B

C

DE

. 85

- ?

. .

. , AB -, CD - (AB CD ), E , CD , A B (. 85)., AE 6 AB BE 6 AB,AEB > 60. , CE > CD DE > CD, CED 6 60. CED > AEB .

6. (.) ABC P . A,B, C P BC, CA, AB. , -

79

A B

C

AB

C

A1

B1

C1

P

Q

. 86

, ABC, ABC.

. A1, B1, C1 , P - BC, CA, AB. CA1 = CP = CB1, - A1B1 A1CB1. A1CB1 = 2ACB, ACB (. 86)., A1C1 B1C1 - ABC. -, Q , A1B1C1, ABC. ABC - A1B1C1 P 1/2, , ABC, PQ, , ABC.

9

1. (.) R. -, R, ., , , .

. O , O1, O2 - , A, B . O1 ,

80 (2006)

AB

C

O

O1

O2

. 87

OB, O2 , - OA. C, AB.

O1C = O1A O2C = O2B

, C - (. 87).

2. (.) , A M . - BC, -M. , , -

ABC, .

. O , O , ABC, P - ABC. ABC P 1/2, P OO 2 : 1. , , M, OM, ABC , A 2/3., O (. 88).

A

B

C

M

P

O

O

. 88

81

, ABC, , - , , O ( M O, ).

3. (.) ABC A1B1C1 -. AA1 A ,

AA

A1A=

BC

B1C1.

B C. , A, B C .

. , ABC A1B1C1, - l , l , k, - . , AA1, BB1, CC1 l , k, . . A,B, C l .

4. (.) 90, 270 . , ?

. t t4 + t2 = 1. ABCDEF ,

AB

BC=

BC

CD=

CD

AF=

AF

FE=

FE

ED=

1t,

AB

C D

EF

. 89

82 (2006)

, . 89. ABCDEF t t2.

5. (. ) , ABC, . .

. 1 : 1.

. , , . , AC, BC X , Y , C J ; d1 J AB, d2 J .

2SCXY = (CX + CY)d2,

2SAXYB = (AX + BY)d2 + AB d1, , d2 = d1, . . J .

O, - I H . . A B. AI , BI HAO, HBO,,

AH

AO=

HI

IO=

BH

BO.

AO = BO, AH = BH , . . ABC 1 : 1.

6. (. , . ) ABCD.A, B, C, D BCD, CDA, DAB, ABC., ABCD ABCD .

. . KLMN ; X , Y -

KL NM a; U , V LM KN b. XY UV b, a (. 90)

. A1, B1, C1, D1 BCD,

CDA, DAB, ABC; A2, B2, C2, D2

83

K

L

M

N

U

V

X

Y

. 90

. A1B1C1D1 - ABCD - 1/3. , - . -, A2B2C2D2.

P ABCD.

AP

CP=

AP

BP

BP

CP=

sinABD sinACBsinBAC sinCBD .

A2B2C2D2 - ABCD (, A2, B2 CD), A2C2.

P1, P2 - A1B1C1D1, A2B2C2D2; P AC, A2P2/P2C2. A1, C1 AA2,CC2 2 : 1, , P1 PP2 2 : 1. - BD, . , P - ABCD, , A1B1C1D1, A2B2C2D2 ABCD.

10

1. () . , , , .

. O . l1 A1 l3 A2, B A1A2 l2 (. 91). OA1B OA2B, ,

OA2OA1

=sinA1OBsinA2OB

.

84 (2006)

O

B

A1

A2

l1 l2

l3

. 91

A2 l5 A3, A2A3 l4 . . ,, A6 A1.

2. (.) X ABCD . Y , X - . , B AX , XC, CY , YA .

. , X ABCD, . K , L, M, N X AB, BC, CD, DA; K , L, M, N , X - . K , L, M, N , K ,L, M, N . BK = BX = BL, K L B - K BL, . . BX B. , , , X ABCD, , X , K LMN . KLMN XX , , X Y . , XKBL, XLCM, XMDN , XNAK,

AXB + CXD = KXA+ KXB + CXM+ DXM =

= KNA+ BLK + CLM+ MND =

= (p KLM) + (p MNK) = p. , XB DX AXC. , YB DY AYC. , , BAD XAY , BCD XCY . ,

85

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L

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N

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. 92

, BA, BX , BC, BY - AXCY , D. , , (. 92).

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86 (2006)

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. 93

4. (. ) , ABC, A1, B1,C1. , A, B, C - , A2, B2, C2. , A1A2, B1B2, C1C2 .

. (.) A - w B C ( B C). , , AA ABC (. . , AA1 - A). AA w A0. A1AB = A0AC, BA1 CA0 .

ABC , w , l BAC. , A1 A0 - . , l BC, A A2 (.93), , , A1A2 AA. , AA, BB,CC L ABC, A1A2, B1B2, C1C2 , L ABC.

, A, B, C , .

5. (.) - 3, 4 5 ( )?

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87

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3. , - n- , n- n . n ? (89 )

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) ? (8 )) ? (810 )

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) , .., , ? (1011 )

7. .

89

( ). , ? (89 )

8. P, - A, B, C . A1, B1, C1 AP, BP, CP .C2 AB1 BA1. A2, B2 -. , A1B1C1 A2B2C2 . (89 )

9. , . . (89 )

10. -, 3 A, B, C (. . ). (89 )

11. . , 1 , , 2 . ? ( 0,1, 6000 .) (810 )

12. ABCD P. , A B , , PC PD, Q. , PQ AB. (910 )

13. AB ABC X , Y , AX = BY . CX CY U V . , UV . (910 )

14. AD BC, P Q - AC BD . , , DAQ = CAB, PBA = DBC. (911 )

15. ABC AA, BB CC. ABCC = P AC BB = Q. , PAC = QAB.(911 )

16. A, B. M AB , A1, B1, A2, B2. A1B2 A2B1 AB P Q. , M PQ. (911 )

90 (2007)

17. - ? (911 )

18. - . (911 )

19. A, a, , B C. , M, AB AC P Q . a SPAQ < SBMC? (1011 )

20. - 1. . . (11 )

21. ( - 4 ). , , . , , . . (11 )

..

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2. . , 15 . 7,5 . , , . 5 . . - 3 .

1 Quantum 78, 1998. .

94

1 . . 6,5 . 6,5 2 13 . 3 . .13+3 = 16 . ? ( - .)

, , . - , . , - . ( .) .

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. ABC . 1. CAB = C1A1B1, AB = A1B1. - ABC A1B1C1 . ( AC = A1C1.) ,

ABC = ABC, AB = AB.

BB. BAB . ,

ABB = ABB.

, CBB = CBB. , CBB CB = CB. ,

A

B

C

B

. 1

... ? 95

ACB ACB , . . ABC A1B1C1 . ?

. , , . : ? . , -, . , , , , , , -. . . , .

4. ABCD, ABD = 40. , , ABC CAD BD. ABC?

. O Q , ABC CAD. AC , OQ - AC . , . , - ABC = 80.

?

5. , p q x2+px+q = 0. p q.

. : p + q = p,pq = q. , : p = q = 0 p = 1,q = 2. - ?

6. tg(x + p/4) = 3 ctg x 1. . -

y = tg x. :

y + 11 y =

3y 1.

y = 3/5. , x = arctg 3/5+ pk. ?

7. log1/16 x = (1/16)x?

96

OA

B

C

. 2

. - . - , , . , - . ?

. - . .

8. - . , . .

, .

9. ., , - , 5 . .

- : S = 2pRh, R , h .

. R r - . , (. 2).

OA = OB = R, AB = r.

... ? 97

B BC OA. AC , , - . AC = h. - ABC OBC :

r2 h2 = R2 (R h)2.

: h = r2/2R. S = pr2. 4pr2. 4 . , 5 . . , (?)

1. 5 . .

2. 13 3 . , , . 13 = 10 + 3, 10 , , 3 .

3. BB C, . CBB CBB , 0, , .

4. : . - . : 90.

5. , p q . p = q, , : p = q = 1/2.

6. , , x = p/2+ pn.

7. , 1/2 1/4 -. , . . - ,

98

. , loga x = ax

. ( : .)

8. , , -. a , g , 0 < g 6 a. sing. , a 6 90, . a > 90, , g = 90. , a > 90 sina = 1/2, a = 150.

9. , 5 ( )

5.

, , , , .

..

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.

1. , - , , .

. - , : a = 2R sinA. (

1 8, 9, 1991.2 15 ,

. ( [email protected]).

100

a, b, c , A, B, C , ha, hb , hc , R .) , - , . , . , .

, . , , , - ( - ). . XXIV - (1990).

.

. AB, BC CA C, A B , -

A B

C

A

B

C

M

. 1

ABC, , - A, B, C; A, B, C A,B, C . ( -, - M .) - - . , - , - - A, B C. -, .1, M ABC ABC, , A, B, C, M .

.

2. AB ABCD E , A B. AC DE F . , , ABC, CDF BDE , .

. . 2 , , 2 ,

... ? 101

AB

C

D

E

F

. 2

AEF ( E B, B C, F C, C B, D A). , 2 , , -, . , 2. , .

. -

x y

z

a

b

c

. 3

, - ( ?). , a, b, c 1/ha, 1/hb , 1/hc . a,b, c, x, y, z. - , x, y, z. , (. 3).

3.

x2 c2 +

y2 c2 = z,

y2 a2 +z2 a2 = x,

z2 b2 +x2 b2 = y.

, , ( 1/a, 1/b, 1/c), a . ( , , , .)

102

, , , . ( , , . - , , .) - , , , . , - ( H) ABC :

HAB = HCB, HBA = HCA, HAC = HBC.

.

4. M, -

MAB = MCB, MBA = MCA,

ABC .

. , - . , : -? . AM,

A B

C

MA1

B1

C1

. 4

... ? 103

BM CM A1, B1 C1 (. 4). A, C, A1 C1 . ,

MA1C1 = MCA = MBC1 MAC = MC1A1.

, M, B, A1 C1

MBA1 = MC1A1 = MAC.

a = MCA, b = MCB g = MAC, , a+ b+ g = p/2, , AA1, BB1 CC1 . - . AB ABC , BC CA ( ).

, .

--, , , . , , . - 1990 , , 1 2 1990 1991 .

, -- ( ) -( , ).

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5. . , - .

6. , , , .

5 6 , .

104

7. - ABCD, M. -

A

B

C

D

M

S1 S2

S3S4

. 5

S , , ABM CDM 1/4.

. -, (. 5),

S1S3 = S2S4 =14

( 5), ,

S = S1 + S2 + S3 + S4 >

> 2S1S3 + 2

S2S4 = 2

( -).

( 6), S 6 2 , , 2. , ABCD 2.

, , . , , , , ( !). , - , , XXIV .

8. A1A2 A2A3 2n-A1A2 . . .A2n K N ,

KAn+2N =p

2n.

, NAn+2 KNA3.

. . ABC B -, A C (.6). , , - ABC. ( , . , .)

... ? 105

P. , APC 90 (1/2)B. : B

A

B

C

P

. 6

ABC P ,

APC = 90 12B,

AP CP . (-.) . , 8.( ABC - KA2N , P An+2.) 2n- , - .

- -. , , , .

8 . , , , , ( , . . 6) , , . , , ...

9. ABC B 120. AC M, AB K , BM ABC, a CK , ACB. MK BC P. , APM = 30.

. . BMC BK CK B C (. 7)., MP BMC, P , B M ABM. ,AP BAC.

APM = PMC PAM = 12(BMC BAM) = 30.

106

A

B

CM

K

P

. 7

, ( - ) , , . - ( ), - . ( ), . , - , , . , .

10. - , 6, - 10, 7, 60. .

. , 60 120 ( 60, !), - , , , , . , - . 6, , , 10. ABCD

AB = BC(= 10), AD = DC(= 6),

O1 O2, (. 8). ,

... ? 107

A

B

C

D

O1

O2

. 8

7/3. -

O1 ABCD , ABCD

16 73. ABCD -

60 ( 60,

A C ), 16 73

> 60. -

, O2, , - ABCD, 7/3.

SABCD = (10 6) 73,

V =196

33.

, - , , ., , . A, B, C, D, E F , , , - AB DE , BC EF , FA CD, . .

11. ABCD , AB CD M, BC AD K ., B D KM.

, , , , .

, , - , , ,

108

. , , . ( .) , , .

. (-, ). , 2n A1, . . . , A2n , ,

A1A2 = A2A3 = . . . = A2nA1.

2n , - A1 , , .

. . , , .

12. R r a. , - , .

, :

R2 + r2 a2.

, , : ) ;) ; ) ; ) . - , . , , , - . , .

, , .. -- , , aha = bhb , - : , - , ? .

... ? 109

13. A B. - C , ABC :

) ama = bmb (ma mb ABC);) ala = blb , (la, lb ABC). . ). AA1 BB1 , AA0 BB0 .

AA0A1 BB0B1. A1, A0, B1, B0 - ABC, . 9. A, B, A0 B0 . A0B0 AB AB0A0B, , AC = BC. C,M, A0, B0. CA0MB0 A0B0 AB CM . CM (2/3)mc A0B0 3 : 1.

m2c =34AB2.

, - AB AB (AB

3)/2.

) - AB, , AB 60.

, - . ( ).

A B

C

A0

A1

B0

B1 M

A B

C

A0A1

B0

B1 MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM

. 9

110

, , ( ). , , , - , . . .

14. ABC, :

) A B,) KA1 KB1, AA1 BB1 -

, K ,) C1 (CC1 C)

CA CB,) C1A1 = C1B1,

A B

C

A1B1

C1

A2

B2

. 10

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-. ) ).

, . - , , - , , .

AA1, BB1 CC1 - ABC (. 10).

AC BC A2 B2 , CA2 = CA1,CB2 = CB1. , A1, A2, B1 B2 . , AC1 BC1 , A B , AB C1. ( , - , . .) , :

AC21 = AB1 AA2, BC21 = BA1 BB2.

... ? 111

ABC (AB1 == bc/(c + a) . .), ,

(a+ b + c) (a+ b)2 = c(c + a) (c + b).

, , - . - . c = 1, a + b = 1 + l. ab = l(2+ l)2. l , a b, a 6= b.

. , ( ) , . , , . , , .

15. , ABCD :

) ,) ,) 180,) BAD = BCD = ABC = ADC,) BAC=BDC, ABD=ACD, BAD=

=BCD,) ,) ,) ,) , , -

,) ,) ,) ,) 2,)

, ?

!.. , . , , , . , , 8 ,

112

, , ) ( ) ) ( 180).

-, . , , -, ) . . - , . , .

). , - .

, ) . , . A, B, C D , .

-. , , , , .

. - - . , .

16. ABCD AD , B C - AD. , AB + CD = AD.

, , . , , AD . . .

16. , B C - ABCD AD, AD = AB + CD.

( ) -. P AD, B C. BCP M -

... ? 113

AD. , ABCD BCPM, ABM (AB = AM) CDM(CD = DM). .

, , -, , , . , . 1990 (.: 10, 1990) .

17. D E AB BC ABC. K M DE . BK BM AC T P. ,

TP 613AC.

. , , . , . , ( ). , - , .

17. , . D E ,

K M. , KM

DE ,

DK = ME . - ,

, E D1, K1 M1 (. 11). DK = ME = a, KM = b,OD1 = lOD. D1 DE , :

D1K2 = la, K2M2 = lb,D1M1M1E

=l(a+ b)

a,

D1M1 =l(a + b)

l(a+ b) + aD1E , D1K1 = la

la+ b + aD1E ,

K1M1 =

(l(a+ b)

l(a+ b) + a la

la+ b + a

)D1E.

114

D E

O

K M

D1

K1

M1

K2M2

. 11

-

l(b2 + 2ab)(l(a+ b) + a) (la+ a+ b)

6b

2a+ b,

(l 1)2a(a+ b) > 0. -

, , . .

18. AB CD,A C , B D . , AC BD , , .

. . ( , - , . .) , , , - . . - . : , , - .

19. , ?

... ? 115

. 12

. -, , - , .

-, . : - . ( , , . - -, , - . . .)

. , . , A1A2 . . .An (. 12, n = 5). - n- A1A

2 . . .An ,

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n .

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n , . . .

, - ( ). - .

116

20. n , . n-, . n- .

- . , . , , - 20, , ?

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1989 . , . -, ! ( . . . , .)

, , , . , - . , .

- .

22. - , ?

: 6. , , , . , , . , l1 l2 , O.

... ? 117

: l1, - l2 ; , l2 ,l1 . .

, , (-). , . : (a, b, 0); (b, 0, a); (0, a, b) - . , - . , a > b > 0, a2 ab b2 = 0.

b = 1, a =1+

5

2.

, , . , : - , , 1. , . , , . , , - , .

- , . - . , , (, , ), - - . , , - (9, 1978), , -. , .

118

. 13

. 14

, - . - : 4+ a 0 < a < 1? , , . , , .

, 4+ a 17 . a, - .

, , - . , ,

... ? 119

a. , . 13, a =

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- 51 (..). , , (!), , a , . 14. ( -, a , . 13).

, - . , - .( , ?) , - . , , , - . ! !

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1996 . . - . . . - 10 , 23. :

124

, , , .

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, - -, 199 -. : - , ., , --. , , , -. , , . , , (! , !), ( , ? ? , !).

, -. , . : . , , , . . ... , . , . , 4 : , , .

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.

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1 1, 2006.

.., .. . .. 139

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AB. CBD 60, B .

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5. : - . ,

6. BA ( ABB) BBC (!).

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8. ABC ABC, . . - 90.

1 ., . -

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140

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: 1? , -

.

, p ( ), , .

. . . , , . 2: B = 120, BB - BBC = 60.

. 5. ( , , - - ). - .

, . : - , - , -p . , , ABB, , . - , , BA . , .

: p, , . , - , , .

.., .. . .. 141

p , .

2. ABCD , - AD, . , AD( AD) AB + CD.

, . , , ( ), , , , . -, .

.1. B, C O, O AD

, AB, BC CD (!).2. M ( , O),

. 1 AD, AMB a (!).3. p MBCO , BCO = a.4. CB CD ( )

O, CO .5. . 4 3 , BCD = 2a.6. p ABCD ( ), ,

. 5 , BAD = 180 2a.7. ABO A O, ,

ABO 180 (180 2a+ a) = a.

8. . 7 , ABO , . .AB = AO.

9. , CD = DM, - .

, 9 , (, B, C O , , , , M ). : . 3 , ; . 4 ; . 6 p; . 7 ; , . 9 .

142

, , .

1993 - ( ) . .. . 9 . .

3. ABCD. ,

BAC = 30, D = 150

, , AB = BD. , AC - C.

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.4. . 3 AB = BD , A, B D

B ( AB) (!).5. . 3 4 , ADB 60.6. . 5 , ADB = 30.7. . 6 , D B C.8. , CD CB, . . AC -

BCD, . . -

, ? , : . 3.

. .

: ? , ,, . . , . ( , ), , -, - , . -, , ,

.., .. . .. 143

. .

.1. BCA f, a DCA y, AB

a, BC b.2. ABC :

a

sinf=

b

sin 30= 2b. (1)

3. CAD q; (ADC = 150), . 1 - :

q = 180 150 y = 30 y.4. (BAC = 30 AB = BC) . 3 ,

A = 60 y = BDA.5. . 4 , BDC = 90 + y.6. CBD :

a

sin(f + y)=

b

sin(90 + y). (2)

7. (1) (2) :

sin(f+ y) = 2 sinf cosy.

8. , :

sinf cosy = cosf siny.

9. . 8 , f = y. . ?

, : - . : , , .

56 , - . - , : , !

, - , - ... 1984 .

144

. . , . - . , : - , , , - . -, ( ). , . : , , . , , , ; 1980- .

4. O, AC , OC F . , BD F , ABCD -?

. . .

1. ABC OBF .2. , ,

, OF AC.3. :

SOBFSABC

=OF

AC=

SODFSADC

.

4. , , ,

SOBDSABCD

=OF

AC.

5. , , -, OBD.

6. , , sinBOD. .

7. BOD f. f. -, f0 6 p, f0 AC

.., .. . .. 145

BD, , f0 6 p/2, f = p/2; f0 > p/2, f = f0. ( .., .)

, - - . .

1. . AC = 1, OF = a, aBFC f.

2. , , - , .

3. BD 21 a2 sin2 f.

4. . 2 3 ,

g(f) =1 a2 sin2 f sinf

, 0 6 f 6 p/2.5. (a sinf)2 = z

f(z) = (1 z)z , 0 6 z 6 a2. ,

: a 6 1/2, z = a2, . e. f = p/2, 1/

2 < a 6 1,

z = 1/2, . . sinf = 1/(a2).

, .

, - , p , . , , -,: - !. - - . .

A M N . BC M , ABNC ?

, , , (.: , 2 ., . 2006). - : .

... , , . .

146

5 ( ). AB, M N , AM = BN . M N PQ RS . QS RP AB K L. , AK = BL.

. , - , . - M = N . , M AB, PQ RS, , AK = BL, K L - AB QS RP . . : 1815 Gentlemans Dairy ( , , ). , . , , -, . , , , . . , , , .

? , , . ,

A BK L

M N

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