1、 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http:/ 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http:/ 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http:/ 1994-2006 China Academic Jour
2、nal Electronic Publishing House. All rights reserved. http:/ 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http:/ 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http:/ 1994-2006 China Academic Journal Electronic Publishing Hous
3、e. All rights reserved. http:/ 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http:/2002 IMO ABCD E F , P, P PO EF O. : BOC= AOD. 1 : 1, OP AOC , DOB . AC EF Q, AEC F, EB BA AQ QC CD DE =1. AEC BPD, ED DC CP PA AB BE =1. AP AQ = PC QC . P EF OA OC I J , PI QO = AP
4、 AQ , JP QO = PC QC . PI QO = JP QO PI = PJ. OP IJ , OP IOJ , OP AOC. : BD EF , OP DOB; BD EF , B ED AC G( 2) , 2 AG AC = AB AE = AD AF . GD CF, , BCDG . , P BD . , OP DOB. a1 = 1 4 , an = 1 4 (1 + an - 1 ) 2 , n 2. , x 1 , x 2 , , x 2 002 , 2 002 k=1 Ak a2 002. A k = x k - k ( x k + + x 2 002 + k(
5、k - 1) 2 +1) 2 , k 1. : ( k) = 1 2 k( k - 1) . . 1 a 0, c 0, b 0, f ( x) = a x + b + x - c ( x + b) 2 . x = (1 - a) b +2c 1 + a , 1 4 (1 + a) 2 b + c . : y = 1 x + b , f ( x) = - ( b + c) y 2 + (1 + a) y = - ( b + c) y - 1 2 1 + a b + c 2 + 1 4 (1 + a) 2 b + c . , y = 1 2 1 + a b + c , x = (1 - a) b
6、 +2c 1 + a , f ( x) max = 1 4 (1 + a) 2 b + c . 2 a 1 = 1 4 , a n = 1 4 (1 + a n - 1 ) 2 , n 2. a n 0 a n 1. 3 n 1, n k=1 A k 1 ( n +1) +1 a n , . : 1, x 1 - 1 ( x1 + + xn +1) 2 1 4 1 x 2 + + x n +2 = a1 x 2 + + x n +2 , x 1 = x 2 + + x n +3 , a 1 x 2 + + x n +2 a1 x 2 + + x n +2 + x2 - 2 ( x 2 + +
7、x n +2) 2 1 4 (1 + a 1 ) 2 x3 + + xn +4 = a 2 x3 + + xn +4 , 7 2 2003 1 x 2 = (1 - a 1 ) ( x 3 + + x n +4) +4 1 + a 1 , a 2 x 3 + + x n +4 . a n - 2 x n - 1 + x n + ( n - 1) +1 + x n - 1 - ( n - 1) ( x n - 1 + x n + ( n - 1) +1) 2 1 4 (1 + a n- 2 ) 2 x n + ( n) +1 = a n- 1 x n + ( n) +1 , x n - 1 =
8、(1 - a n - 2 ) ( x n + ( n - 1) +1) +2( n - 1) 1 + a n - 2 , an - 1 x n + ( n) +1 . a n - 1 x n + ( n) +1 + x n - n ( x n + ( n) +1) 2 1 4 (1 + an - 1 ) 2 ( n +1) +1 = an ( n +1) +1 , n x n = (1 - an - 1 ) ( ( n) +1) +2n 1 + a n- 1 , a n ( n +1) +1 . , , , n , n k=1 A k 1 ( n +1) +1 a n , x n = (1 -
9、 a n - 1 ) ( ( n) +1) +2n 1 + an- 1 , xn- 1 = (1 - a n- 2 ) ( x n + ( n - 1) +1) +2( n - 1) 1 + a n - 2 , , x 2 = (1 - a 1 ) ( x 3 + + x n +4) +4 1 + a1 , x1 = x2 + + xn +3 , . 3, = 1 (2 003) +1 = 1 2 003 1 001 +1 . 17 , 17 . : (i) ; (ii) , ; (iii) 6 . ? . : 17 17 ,17 17 ,17 17 . i j , i j . , 4 4 ,
10、 6 , . 1 6 . 17 17 17 6 17 11 . 1 6 , 16 1 . , 22 . 2 1 , 16 11 . i x i , . , i C 2 x i ( C 2 1 = C 2 0 = 0) . , C 2 x 1 + C 2 x 2 + + C 2 x 16 C 2 11 =55. , x1 , x2 , , x16 ( 1) , , , x 1 + x 2 + + x 16 . , x1 , x2 , , x16 4 14 3 , C 2 x 1 + C 2 x 2 + + C 2 x 16 =54, x 1 + x 2 + + x 16 =50. , 51 ,
11、. 17 17 72 . , 71 , 1 6 , 4 4 . 71. , 72 . 72 . , 6 22 , 11 50 . 50 16 11 : (1)2 4 , 14 3 ; (2)3 4 1 2 , 12 3 8 2 . (1) . 17 17 , , 1 6 1 2 3 4 5 6, 2 5 1 7 8 9 10. 3 5 , 14 4 . 7 8 9 10 4 . , 15 4 1 . 1 , 7 2 3( 1 3) . 4 7 7 , 15 3 , 4 17 4 16 . 2 10 , 22 + 16 = 38 , 15 7 34 , 34 =3 4 +2 11. 3 4 +1
12、1 =23 ,23 21 = C 2 7 , , . (2) . 1 6 1 2 3 4 5 6, 2 5 1 7 8 9 10. 3 4 5 , 3 , 12 4 . 7 8 9 10 . 16 , (1) . , 3 , 6 . 7 8 9 10 , ; 7 8 9 10 1 , 7 , 7 8 9 10 17 4 17 , 7 15 7 33 , 1 . , 14 7 32 . 32 =3 4 +2 10, 3 4 +10 =22 21 =C 2 7 . . ,17 71 . (a) n( n 2) , a 1 , a 2 , , a n , | a i - a j | | 1 i n,
13、 i 1. i , b i +1 =2. , b i n - 2. , b i = n - 1. b 3 = 2, b 4 = n - 2. , b 1 + b 2 = b 3 + b 4 . . , n - 1 4 , n 5 a 1 , , a n . bn - 1 =1, bn- 2 = n - 1. n 5 a 1 , , a n . : 0 = a1 a2 an , a n = n( n - 1) 2 . , i j, | ai - aj| = a n - 1. , a n - 1 = a n - 1 - a 1 = a n - 1 a n - 1 = a n - a 2 , a2 =1. 9 2 2003 1
