1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

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1 0 9 ( ) 10 (2000 ) SAT ACT 1

2 1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 3, 1 2, ABCD AB, BC, CD, DA 5 : 2, P, Q, R, S AQ BR W, BR CS X, CS DP Y, DP AQ Z W XY Z 6 2, 3, 5 90,,,, n ( 2 ) n 8 f { n 3, n 1000 f(n) = f(f(n + 7)), n < 1000, f(90) 2

3 9 n, a n = 50 + n 2 n, a n a n+1 d n n ( ), d n 10 x, y, z x + y + z = 1, 1 x + 4 y + 9 z 11 ( ), 3 ( ) 12, , {a n } a 1 = 1, n 1 a n+1 = a n + 1, a 100 [a 100 ] a n 3

4 1990 IMO 1990/2/11 1 (xyz ) E A 1, A 2, A 3, A 4, A 5 E, (1), (2) (1) A 1 A 2 A 3 A 4 A 5 = E (2) i j A i A j = φ, A 1, A 2, A 3, A 4, A 5 2 n 3, a 0, a 1,, a n 1 a 0 < a 1 < < a n 2n 3 a i + a j = a k + a l = a m i, j, k, l, m 3 X, (1), (2) (1) x X 4x X (2) x X [ x ] X, [a] a, X 4 n n a 1, a 2,, a n K < a 1 + a a n < G a 1 + a 2 a 2 + a 3 a n + a 1 K G 5 A, B n 2n P (n) P (n), 2n k, k A, k B Q(n) ( ) Q(8) ( ) Q(n) 4

5 /1/15 1:00-4:00 1 A = (81 9) A 2 2 x x 5 = 0 (199 ) ABC G GA = 2 3, GB = 2 2, GC = 2 ABC 4 1 x y + 1 (x + 1)y = (x, y) 5 8 (x, y, z) (x, y, z 0 6) P (e, π, 5) P (6 ) [ n ] [ n ] 6 n, f(0) = 0, f(1) = 1, f(n) = f( ) + n f(n) 0 n 1991 f(n) [x], x 7 n = n 5 n 8 n = 2 i 3 j 5 k (i, j, k ), 10 4 < n r n 63 (n, r), n! nc r = r!(n r)! 0C 0 =

6 11 A, B 15 : ( ) AA 5, AB, BA, BB 3 AABBAAAABAABBBB, AA 5, AB 3, BA 2, BB 4, ( 12 ) 6

7 /2/11 1 ABC BC, CA, AB t : (1 t) P, Q, R AP, BQ, CR K, K ABC L L (t ) 2 N N N p, q p(1) = 2, p(2) = 3, p(3) = 4, p(4) = 1, n 5 p(n) = n q(1) = 3, q(2) = 4, q(3) = 2, q(4) = 1, n 5 q(n) = n (1) N N f, n N f(f(n)) = p(n) + 2 f (2) N N f, n N f(f(n)) = q(n) A 16 A, A 4, 4 10, 14, 1 140,, 0 1 :, A n (n 2) [ A n ] ( ), A 3, [x] x 7

8 /1/15 1:00-4:00 1 {a n }, a 0 = 1, a 1 = 2, a n+2 = a n + (a n+1 ) 2 a x 2 + x + 1 = 0 ω ω 2k (ω + 1) 2k = k 3 y 2 = x x 8019 E 2 (3, 9), (4, 53), E x 4 A 1), 2) 1) 2, 3, 5, 7, 11, 13 2) 2 2, 3 2, 5 2, 7 2, 11 2, , 1 A A n n ,,,,, 7 6 ABC, BC, CA, AB 3 : (n 3) D, E, F (, n > 6) AD, BE, CF 4, 49, n 7 x, y, x 4 + y 4 x + y < m 1/3 n < 10 3 m, n, n n 8

9 9 A, B A = { (x, y) x, y 1 x 20, 1 y 20 } B = { (x, y) x, y 2 x 19, 2 y 19 } A, 219, 180 B, (1, 1), (1, 20), (20, 1), (20, 20) 2,, : 2, 2, 2 ( 1 ) 237, ( 1 ) 10 n n, ABC S T, S 441, T 440, AC + CB A C S B A T C B 12 A = { 1, 2,, 10 } A A f, 1) x A, f 30 (x) = x 2) k, 1 k 29, f k (a) a a A, x A, f 1 (x) = f(x), f 2 (x) = f(f 1 (x)),, f k+1 (x) = f(f k (x)), 9

10 /2/11 1 x y, xy 1, n, x + y x n + y n 2 1 ABC AB, AC D, E, BE, CD P BCED P BC 2 D, E AB, AC, P DE 3 n 2, n 1 k=1 n n k 1 2 k 1 < 4 4 A (m, n) 1) m n 2) 0 1 3) f { 1,, m } { 1,, n }, (i,f(i)) 0 1 i m, S { 1,, m } T { 1,, n } (1) i S, j T (i, j) 0 (2) (S ) + (T ) > n, f 1 i 1 i 2 m f(i 1 ) f(i 2 ) 5 n 2, a 1, a 2, a 3, a 4 : i) i = 1, 2, 3, 4, n a i ii) k = 1,, n 1 (ka 1 ) n + (ka 2 ) n + (ka 3 ) n + (ka 4 ) n = 2n 10

11 , (a 1 ) n, (a 2 ) n, (a 3 ) n, (a 4 ) n, n 2,, (a 1 ) n + (a j ) n = n j, 2 j 4, a (a) n, a n 11

12 /1/15 1:00-4:00 1 n , 120 n 2 12, 1, 2 6 2, 1, 3 1 4,, 4 3 (0, 0), (276, 153), (a, b) 2, a, b (a, b), 5 1 ABCD P, Q, AP + BP + P Q + CQ + DQ 6 A = {1, 2, 3, 4, 5, 6, 7} A f, (1),(2), (1) j A, j = f(k) k A, k j (2) A A g, j A, f(j) = g(g(j)) 7 3, A,B 2 9 A 6 3, 3 9 8, A A = { (x, y, z) 2xy z 2, x + y 1, x 0, y 0 }, B B = { (u, v, w) A (x, y, z), 0 ux+vy +wz 1 }, B, A 9 S = {1, 11, 31, 51, 71}, {a n } (1),(2),(3) (1) a 1 S, 12

13 (2) a n+1 1 a n + 1 S, (3) 10 n a n = 1993 {a n }, a ,,, 11, , 3,

14 /2/11 1 a, b, c,, x, y, z, 2, kyonkyon 8 2, 2 W 1 W 2, 1, 1, W 1 W 2 2 n, n d(n), D(n) T (n) : D(n) = d(1) + d(2) + + d(n) T (n) = n, 3D(n) = 2T (n) n 3 x, y,, 2 2 3, x, y, x, y,,,, ( ) ( ) 4 S 5 l 1,,l 5, 3 l 1,,l , 5 5 (n a 1,, a n ) C : n a 1,,a n, n n max x a j C n max x a j 0 x 2 j=1 0 x 1 j=1, max f(x) α x β f(x), b j α x β b 1,, b n b 1 b n n j=1 14

15 /1/15 1:00-4:00 1 (x, y), x, y y = 3 7 x a = 2 + 3, 2 a 3 ABCD-EF GH AF H BDE θ (0 θ 90 ) cos θ D C A B H G E F 4 P P (a, b) a + b 4 0, 1, 2, 3, P,,, 1 P 0 10 (0, 10) P 0 5 ABC AB, AC D, E, BE, CD P ADE, BP D, CEP 5, 8, 3, ABC , 15

16 7 5 5,, 8 A = {0, 1, 2, 3, 4, 5, 6, 7}, (1), (2) A A f (1) i, j A, i j f(i) f(j) (2) i, j A, i + j = 7 f(i) + f(j) = 7 9 a > b a, b x n = a 2 n 2 + 2bn x, {x} x (0 {x} < 1), lim { x n } n, lim a n n a n n 10 A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, (1), (2) A S (1) S 5 (2) S, 1, x, y 1 f(x, y) { f(x, y) + f(y, x) = 0 f(x, x + y) + f(y, x + y) = km, 1km 11, 11 y = m (m = 5, 4,, 4, 5), x = n (n = 5, 4,, 4, 5) 5 A k (k = 1, 2,, 5),, (x k, y k ) ( 5, 13), (2, 45), (44, 3), (4, 1), ( 27, 2) 1, S(x, y) (x, y) 16

17 /2/11 1 n, n a n b n = n+a n, b n (n = 1, 2, ), {c n } c n n , A 0 A 1 A 2 P 0, P 1,, P 6, : i = 0, 1,, 5, i 3 k, P i P i+1 A k, (1) P 0 = P 6 (2) : i = 0, 1,, 5, P i P i+1 A 0 A 1 A 2 P 0 4 ABC BC M MAC = 15 B 5 N 1 N, N 1 N 2, A, 1 2 A B A B A, 1 1 2,, 2,, 17

18 a = 3 ( ) a 1995/1/15 1:00-4:00 3, x 3 x 2,,, 1cm,,, 1cm 1cm 1cm 1cm 3 ABC O, OA, BC M, N B = 4 OMN, C = 6 OMN OMN 4 x 2 3x + 3 = 0 x = α α n, k α 1995 = kα n n k 5 (1), (2), p, q x : (1) x q, 1 p 1 (2) x p, 1 2 p q x p 10 x p 18

19 6 4 8, 7 5 A, B, C, D, E, x 2 y 2 + y 2 = 26x (x, y) 9 m m a 1, a 2,, a m, a i 1 4, : a i = a j a i+1 = a j+1 i = j a 1, a 2,,a m m 10, S S = {(x, y, z) x 2 4y 2 + z 2 12xy = 20} 2x + 3y + z = 3, S 11 4, 12 f(x, y, z) x, y, z, x 4, f(x, y, z) { f(x, z 2, y) + f(x, y 2, z) = 0, f(z 3, y, x) + f(x 3, y, z) = 0 19

20 1995 i=1 1995/2/11 1 n 2, r n g n r n 1 ri = 1 (n g), n 2 x x, x x 2 x f(x) a {f(x)} 2 a = f(x 2 ) a f(x) x, x 3 5 ABCDE, AC, AD BE S, R, CA, CE BD T, P CE, AD Q ASR, BT S, CP T, DQP, ERQ 1, (1) 5 P QRST (2) 5 ABCDE 4 {a 1, a 2, } a 2n = a n, a 2n+1 = ( 1) n, P (1) P 0, P P 0 x 1 P 1 (2) P i P, a i , P i+1, i = 1, 2, P 5 k n 1 k n, a 1, a 2,, a k a 1 + a a k = n, a a a 2 k = n, a k 1 + a k a k k = n 20

21 (x + a 1 )(x + a 2 ) (x + a k ) = x k + n C 1 x k 1 + n C 2 x k n C k i (i 1) (i j + 1), i C j 2, j (j 1)

22 /1/15 1:00-4:00 1 xyz- 4 (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) n a n = 10 2n 10 n +1 2 a n 4 a x 3 x 1 = 0, a 2 5 A = { 1, 2, 3, 4, 5, 6 } f: A A, f f f f f 6 N f: N N (1), (2), (3) (1) f(xy) = f(x) + f(y) 1 x, y (2) f(x) = 1 x (3) f(30) = 4 f(14400) 7 (a, b) LCM(a, b) + GCD(a, b) + a + b = ab a b LCM(a, b), GCD(a, b) a b, 8 0 f, f(0) = 0 [ x f(x) = f( ) + 10] [ ] 10 log 10 x 10 [ ] x x 1996 f(x) x 22

23 , x [x] x 9 xyz- 2 ( 2, 0, 1) (2, 0, 1) 4 T, xy 2 B T B, z = 1 2, xyz- A, A 2 P, Q, P Q A 10 n, S = { 1, 2,, n } S ( ) A, B, C, D A B C D = S A B C = φ (A, B, C, D) , 1, 2,, 1996, OFF k, P k, k ON/OFF P k, k = 1, 2,, 1996, ON 12 n 2, 4n (i, j) (i = 1, 2,, n j = 1, 2, 3, 4) L n L n (1, 1), (1, 1) a n : (2, 1) : (1, 1) L n : x y a 2 = 1, a 3 = 2 a 12 23

24 /2/ ABC P QR, ABC P QR, 0, T,, T, ABC, T A, B, C, ABC A, B, C θ ABC T, T, ABC, θ 2 GCD(m, n) = 1 m,n GCD(5 m + 7 m, 5 n + 7 n ) GCD(m, n) m n 3 x x > 1 a n = [x n+1 ] x[x n ] (n = 1,2,3, ) {a n }, n a p+n = a n p [x] x 4 Γ θ Γ θ 5 q < q < 2 n 2 2 n = 2 k + a k 1 2 k a a 0 ( a i = 0 1) p n p n = q k + a k 1 q k a 1 q + a 0, k ( ): p 2k < p l < p 2k+1 l 24

25 /1/15 1:00-4: !, 0? 2 30,, 3 xyz- xy- 13, yz- 6, zx- 18, 4 A = {1, 2, 3, 4, 5 } f: A A ( ) f(f(f(x))) = x x A 5 ABC BC = 6, CA = 5, AB = 4 AB, AC D, E, ADE BC D, E AB, AC, DE 6 a 3 a 1 = 0, a + 2, n, n 0, n 1,, k + 1 n k (0 k 9) n 8 f(x) 5, 5 f(x) + 1 = 0 x = 1 3, f(x) 1 = 0 x = 1 3 f(x) 9 f(x) x,, (1) (4) (1) 0 f(x) 1996 (x ) (2) f(x ) = f(x) (x ) (3) f(xy) f(x)f(y) mod 1997 (x, y ) 25

26 (4) f(2) = 999 f(x),, f(x) 1000 mod 1997 x a b mod n, a, b n 10 1 n, 2,,, 1 n = 5 2, 4, 3, 5 1, n = 7 2, 4, 6, 5, 1, 7 3 n = ABCD, AB = CD = a, AC = BD = b, AD = BC = c, 12 n ( ) 1 i, j 19 (i, j), 1, 2,, n, ABCD, A C, B D,, (10, 10) (, 1,, n, ) 26

27 :00 17: , 10, 2 2 a, b, c, (b + c a) 2 (c + a b)2 (a + b c)2 (b + c) a2 (c + a) b2 (a + b) 2 + c P 1,, P n, 2 ( ) , G 9, : G 5, 5 2 G 4 A, B, C, D 4,,, 3 AX + BX + CX + DX A, B, C, D X = X 0, AX 0 B = CX 0 D 5 2 n A B n n 27

28 /1/15 1:00-4:00 1 A, B, 10 A, B A, B A, B, 8 B n n AB CD ABCD, AB = BC = DA = 1, CD = AD E : E, A DC DE 4, 10 1, 2,, 5 xy- 4 A : (3, 0), B : (3, 2), C : (0, 2), D : (0, 0) ABCD uv- (u, v), ABCD (x, y) 0 ux + vy 1 (u, v) S S 6 8, 2 4, 2 4, 8 7 a n = n 1 (1 n 100) a 1,, a 100, 1 8 n,, , 1998 n 28

29 9 ABC B, C AC, AB D, E, ABC : BDE : CED = 2 : 3 : 4 A x 3 y 2 z 10 x, y, z x 6 + y 6 + z 6 n 11 a 1, a 2,, a n m = a i, i=1 1 n i a i, n,, i a i ( ) 1) 2) k, k 3), 2),, m, a 1,, a n (, ) 12 a n = n 3 5n 2 + 6n, b n = n (n = 1, 2, 3, ), a n b n d n a n = 0 d n = b n d 1, d 2, d 3, d d n = d n 29

30 :00 17: p 3 p, 1, 2, 2 3,, p 1 p 1 p,, , A, B, C, AB, BC, CA, 3 P 1, P 2,, P n P 1 P 2 P n P 1,,, 3 1 P 1, P 2,, P n, P 1, P i, P i 180 P i 1 P i P i+1, P i, P i (180 P i 1 P i P i+1 ) ( (180 P i 1 P i P i+1 ) < 0) P 0 = P n, P n+1 = P 1 720, 4 (1, 2,, n) A = (a 1, a 2,, a n ) n, n, 1, 2,, n, 1 k n, k a k, A = (a 1, a 2,, a n ), 1, 2, n A, A,, f(a) 0 m, f(a) = m n 30

31 A c n,m, P n (t) = c n,m t m m=0 Q n (t) = 1(1 + t)(1 + t + t 2 ) (1 + t + t t n 1 ) P n (t) = Q n (t) ,,, ( ):, 0, 1, 12, 11,, 31

32 , 50, 100 (0 ), (X, Y ) 3x + 5y = 7, X + Y x, y n 1999 n, n 3 n 3 3 n ( ) n = 1990, = 7, 880, 599, 000 = = ABCD-EF GH, AG, 5 1, 1, 2, 3, 2, 4, 5 1, 6 0 n, k p n (k) p n(n k) p n (n k) (0 k n) 6 3 AB = 4, BC = 6, AC = 5 ABC BC P, P 2 AB, AC M, N M, N P P 0 BP 0 32

33 A M N B P C ! 10 n n, 1999! 10 n 8 ABC, A = 60, B = 20 1, AB = 1, BC AC E F y 60 C x A y x B abc + abd + acd + bcd 1 9 n = a b c abcd d > 1 (a, b, c, d), a n, i = 1, α = cos 2π n + i sin 2π n m 33

34 1 m n, 1 n 1 k=0 α mk x α k 12 n ( 3) (1), (2), (3), (1) A, B A B, B A (2) A B, B A A, B (3) C,, C C 34

35 :00 17: ,, f(x) = x n, f(x) 3 n, 3 n+1 x 3 2n + 1, 2n 4 f(x) = (x )(x )(x ) (x 2 + n 2 ) ABCDEF, max{ AD, BE, CF }, min{ AD, BE, CF } 35

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