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. ..
: .. , .. ,..
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
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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
3S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S13S1
4S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S14S1
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
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A2
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. 10
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. 60, 120.
. , B C ,
A
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. 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).
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
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
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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
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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
A
B
C
D
K
L
M
N
K
L
M
N
X
Y
. 92
, BA, BX , BC, BY - AXCY , D. , , (. 92).
3. (.) P , - . , P. .
. X . - X XP . , - P, , , . , . , ,. . X XP, X P . , , . . P .
86 (2006)
A
A0
A1A2
A
B
C
. 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 ( )?
. .
. , , - . 94 ( , - 4 : 1). ,
87
. 94
, , ; , , , .
6. (.) -, . , , - . .
. .
1. , - O - I , L, OI I .
2. - , ().
, M , , - .
1 , - OL. 2, M , - I OL. , M , IM,
X
O LI
. 95
OI , OL, -, L. , OL MI , - -. , -, - OI , , - X XI OXL(. 95). - - - .
(2007)
1. ( ) -. ( ), . . (8 )
2. - . , ?(8 )
3. , - n- , n- n . n ? (89 )
4. , ? (8 )
5. n- , , . n
) ? (8 )) ? (810 )
6. ) ,. . , ? ( ) (89 )
) , .., , ? (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 )
..
?1
, -, , -, , . - . ,, , .
1. 10 . , . 99%. , 1%. ( 98%.) , ?
, , , - . ( .) , -, . , , .
2. . , 15 . 7,5 . , , . 5 . . - 3 .
1 Quantum 78, 1998. .
94
1 . . 6,5 . 6,5 2 13 . 3 . .13+3 = 16 . ? ( - .)
, , . - , . , - . ( .) .
3. . - ABC A1B1C1 : AB = A1B1,AC = A1C1, ABC = A1B1C1, .
. 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.
, , , , .
..
?1
, . -, , , - , , , , - . -, - ( -, ) 2. . , , . , : , . , , - , , . , - . , .
.
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 .
, -- ( ) -( , ).
, . , - ( ) -.
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
) , A1, B1 C1 AB ?
-. ) ).
, . - , , - , , .
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 ,
n-, . M A1 . . .An A1 . . .A
n .
- n- , n-- . , M A1 . . .An M A1 . . .A
n , . . .
, - ( ). - .
116
20. n , . n-, . n- .
- . , . , , - 20, , ?
-. , , - . - , -. , , - .
21. ?
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 =
2/2.
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