2

rev 2004/1/12 KAIST

2

6 7 1 13 11 13 111 13 112 18 113 19 114 21 12 24 121 24 122 26 13 28 131 28 132 30 133 (recurrence) 34 134 35

4 2 39 21 39 211 39 212 40 22 42 221, 42 222 43 223, 45 224 46 225, 48 23 49 231, 49 232 52 24 54 241 54 242, 56 25 58 251 58 252 60 26 64 261 64 262 65 263 68

5 3 75 31 75 311 75 312 76 32 78 321 78 322 81 323 84 33 87 331 87 332 89 34 91 341 91 342 94 35 97 351 97 352 99 4 105 5 109

6,,,,,,,??,,, 2003 9, KAIST

7,,,,?,!,?,??,!!,,,? 1 (1989 IMO )5 5 25, ( )

8,,, 25 @ @, 21? 5 5? 4 4, 6 6, 7 3? 3 3 @ @ n 3 2 2 1 1 4 4?, 12 @ 1 2 @ 12 1, 2 1 8, 8

9 @ @@ @ @@, 8 8 0 2, 8 1 4, 1 2 2 2 1 2 2 1 2 2,, @ @@ @ 1111 1411 @ @ 1111 1111, 3, 2 5 5

10,5 5 4 4 4, 4 4 5 5, @ @ @ 11111 @@ 11111 @ 114 11 @ @ @@ 11111 @@ 11111 6 6 4 4,5 5 IMO?,? 200 4,?,,?, MathLetter

11, BGIVESF work Backward, Generalize, Items, Viewpoint, Extreme cases, Small problems, Fixed point, Induction, Pigeon-hole, Equivalence?

12

1,,,, 11,,?,, 111?,

14,,,,,,?,? 1 a, b φ(ab) =φ(a)φ(b), φ(n) n n 1 ab ab 1 2 3 b b +1 b +2 b +3 2b 2b +1 2b +2 2b +3 3b (a 1)b +1 (a 1)b +2 (a 1)b +3 ab, qb + r q, r r b b, ab φ(b) b φ(b)

11 15 C r r qb + r (q =0, 1,,a 1), b a qb + r q b + r qb q b q q (mod a) q = q, C r a, a, C r a φ(a) C r φ(b), φ(ab) =φ(a)φ(b), 2 (1981 ) 1,, 2, 3, 4? ( ) a, b, c A a 3 3 1, 4 A a, 3 a b 4, 5 A 2 3 A 2, A 5

16 4 1, 6 5 4 5 6 4 5, 6, 5 7 a, b, c 2, a, b, c, 4 4 12, K 5 10 4,, 3 (1988 IMO ) a, x 1988 k=0 1 x k = a f(x) x =0, 1,,1988, (, 0), (0, 1), (1, 2),, (1987, 1988), (1988, )

11 17 1 x k x,, f x ± f(x) 0, x k + f(x), x k f(x), x 0 0 + 1 1987 + 1988 1988 + f(x) 0 0, f(x) =a a =0 1987, 1988, 1 (1990 KMO ) p, q, r A, B, C (1 p<q<r) 3, A, B, C 20, 10, 9 B r, q? 2 (PSTP 123, 1990 KMO 2 ) a b, a b + 2a + b 3a + + b (b 1)a = b (a 1)(b 1) 2 3 (1985 ) a 1,a 2,a 3, m 1, b m =min{ n : a n m }, b m a n m n a 19 =85, a 1 + a 2 + + a 19 + b 1 + b 2 + + b 85

18 112! 4 (2002 KMO ) ABCD, CD M, AD E BEM = MED AM BE P, PE BP AM, EM BC R, Q, ABCD 4a MED = MQC ( ) BEQ M BM EQ BMQ BCM MCQ MC : CQ = BC : CM =2:1 CQ = a = DE PBR PEA PE BP = AE BR = 3a 8a = 3 8 5 (PSTP 121, 1935 10 American Mathematical Monthly) AB

11 19 AB PY P PQ PP Q = PYX, XY Z, XY Z, 1 ABC O A BC D, D BC AO E ADE, B = C 2 (2001 KMO ) ABC = CDE =90 ABCDE AE M, AB CD = BC DE MB = MD 113 x x,, 6

20 S T PQR PQ 3 RS RT 6 8, PQ S, T PR, QR 3 PR, QR 1/3 x, y (2x) 2 + y 2 =6 2, x 2 +(2y) 2 =8 2 5 x 2 +y 2 =20, PQ 3 20 = 6 5, 7 (1976 )8 7, 1, 05, 0 7 8, 8 8?? 8 n, k 1 n +2 (n +2)(n +1) kn +8= = 2 2, n 2 +(3 2k)n =14 n n 14, n =1, 2, 7, 14 n =1, 2 k<0 n =7 k =4, n =14 k =8 (n, k) =(7, 4) 9, (n, k) =(14, 8) 8 7 1 1,

11 21,, 1 (1989 IMO ; ), 38 55, 50 55? 2 (1980 ), 3 A, B, C A, B, C A 1, B 1, C 1, A, B A 2, B 2 C A 1, B 1, C 1, A 2, B 2 114 8 8 8 2 1? 2 1,,,,,

22 9 (Fermat ) p a p, a p 1 1 (mod p), p a 2 a p a p, p a p a,, a p a p p a p a p 1 1 p, a p a p 1 1(modp) 10 (2002 KAIST )1,2,3,, = = (

11 23!), (, ) (, ), (, ), (, ) (, ),,, 1 3 3 n 2 (PSS 2 23 ) A =(0, 0), B =(0, 1), C =(1, 0) A, B, C, D =(1, 1)?

24 12 121,? 1 (1999 ) n, p 1+np n +1 p 1+np = t 2 np =(t +1)(t 1), p (t +1)(t 1) p p t +1 p t 1 t = rp ± 1 n +1= (rp ± 1)2 1 +1=r 2 p ± 2r +1=(p 1)r 2 +(r ± 1) 2 p p 2 (1970 ) y = x y =2x, 4 (a, a), (b, 2b) 4, (x, y) (b a) 2 +(2b a) 2 =4 2 ( ) x = a + b 2, y = a +2b 2 a b x, y a =4x 2y, b =2y 2x

12 25 ( ) x y (3x 2y) 2 +(4x 3y) 2 =2 2 25x 2 36xy +13y 2 =4 3 (1998 Senior ) n : x y (a) a + n a =1 a (a <n), n x y (b) n, a + n a =1 a (a <n) (p, 0) (0,q) q p q ( ) p q p q p = q p (x, y) =(p p,q ) ( ) ( ), a n a gcd(a, n a) =gcd(a, n) a n, n x a + y n a y =1 n a = x a 1,,

26? 2 n,3n +1 n +1 3 (1970 ) f(n) n : 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, f(s + t) f(s t) =st, s, t s >t 122 4 1 2, a b (a = b ) ab + a + b a, b c c +1=ab + a + b +1=(a +1)(b +1), W 1 W, f : W W, f(a) =a +1, f(c) =f(a)f(b) W 2 3, a, b a b, W 2 3 2 p 3 q, W 2 p 3 q 1 (p, q 0 )

12 27 1 (MathLetter 128 Proposal) X (1) 1 X (2) u, v X = uv u + v X X 2 (2002 KMO ) x 3 +2y 3 +4z 3 +8xyz =0

28 13 131, p q p q, p q 1 (2002 ) α =00110101000101000101, α n n (prime number) 1, 0 1 α, 1 m, a(> 1) a + mt (t N) t = a a + mt = a(1 + m), α 2, m C C(2m +1)!+2,C(2m +1)!+3,, C(2m +1)!+(2m +1) k 2m, C 0 a 1923 Eötvös :,,,

13 29 2 (1999 KMO )3 3 3 1, 2, 3,, 27, 3?,,,,, m 27 9 9m =1+2+ +27= 27 28 2, m =42 9 @ @@@ c, c 4 c, 4m =3m +3c, c =14, 14,, 1, 2,,9 3 3 5,,, 1 (1911 Eötvös 1 ) a, b, c, A, B, C ac 2bB + ca =0 ac b 2 > 0

30, AC B 2 0 2 (1973 ) ( ) 132 S N S = N (1) 1 S (2) n S = n +1 S 3 (1989 KMO, ) n ( ), n =1, 2 n = k n = k +1 k +1 k a 1 a 2 a k a a a 1 a 1 a a a 2, a 2 a 2, a a 1,,a j a j+1 a j+1 a a k, k +1 n

13 31, S N S = N (1) 1 S (2) 1,,n S = n +1 S 4 (1991 KMO ) n 1 1 n ( : 2 + 1 2 =1 2+2=4 ) 33 73 33 n 2n +2 1= 1 + 1 + + 1, a 1 + a 2 + + a k = n a 1 a 2 a k 1= 1 2 + 1 2 = 1 2a 1 + + 1 2a k + 1 2, 2a 1 + +2a k +2=2n +2 1 2 1 2 = 1 3 + 1 6, 3+6=9, n 2n +9,33 73 + 2k (k 0) 73 + (2k + 1) = 2(36 + k)+2 73 + (2k + 2) = 2(33 + k)+9 73 + 2(k +1),33 73 33,

32 (Backward induction) S N S = N (1) S (2) n +1 S = n S 5 a 1,,a n - a 1 + + a n n n a 1 a n 1 n =1, n =2 n =4 n =2 a 1 + a 2 + a 3 + a 4 4 a 1 + a 2 + a 3 + a 4 = 2 2 2 a1 a 2 + a 3 a 4 2 a1 a 2 a3 a 4 = 4 a 1 a 2 a 3 a 4 n =4 n =8, n =2 k a 1 + + a 2 k+1 2 k+1 = 2 a 1 + + a 2 k 2 k + a 2 k +1 + + a 2 k+1 2 k 2 k a1 a 2 k + 2k a2 k +1 a 2 k+1 2 2 k a1 a 2 k 2k a2 k +1 a 2 k+1 = 2k+1 a1 a 2 k+1 n =2 k+1, n =2 k - 2 n 2 k n 2 k t n + r = t (r > 0) a 1,,a n m, a n+1 = = a t = m a 1,,a t m t m = a 1 + + a n + rm t a1 a n m t r

13 33, m t a 1 a n m r m n a 1 a n m n a 1 a n, n, n - - n S 1 S, 2 r =1,, 1 (2002 ) S 2002 N 0 N 2 2002 S, (a) (b) (c) N 2 (2003 ) S n, S 1 S n/7, 3 S T 3 a 1,1,,a 1,n ; a 2,1,,a 2,n ; ; a m,1,,a m,n, Cauchy m m n n m a m i,j a i,j i=1 j=1 j=1 i=1

34 133 (recurrence) 6 (1990 3 ) A, B, C,1 2 A,3 4 B,5 6 C,, A B,C,? 7 a 1,a 2,,a n 1, 2,,n a i = i 1 i n n u n, u n 1 (2002 KMO) n 2n +2 x 0,x 1,,x n,y 0,y 1,,y n,, n +1 x 0 x @ 1 x @ @ 2 x n 1 x @ @@ @ @ @ @ n @ @ y 0 y 1 y 2 y n 1 y n 2 (Yaglom MCwES 45 ) 3 n,? 3 A, B A, B A m, B n ( )?,

13 35 134 n n, 2, rn r 8 6,, (3 ) A A B F B F C E C E D D A, B, C, D, E, F, A 3 3,, 3,, B, C, D

36, B C, C D, B D,,, 9 (1976 ) 4 7 (, ) [ ] 10 (1990 KMO) n A 1,A 2,,A n 30, A i, A j, A 1 A 2 A n = φ( ) n<872 1 (2002 ) S {1, 2,,9}, S {1, 2, 3, 5} {1, 2, 3, 4, 5} 1+4=2+3 S? 2 (Crux 1990?) M = { (x, y) 1 x 12, 1 y 13, and x, y Z } (1) M 49, x, y

13 37 (2) 48 (1) 3 (1983 ) n, 1/n n (n +1)/2

38

2 21 211 1 (1926 Eötvös 3 ) k k k k k 2 (1987 KMO) f 1 (x) =(x 2) 2, f 2 (x) =(f 1 (x) 2) 2,, f n (x) =(f n 1 (x) 2) 2,, f n (x) a n b n 1 (1973, 1976, 1988 KMO) O O AB XY, AX BY P

40 2 (2002 ) {P n (x)} : P 1 (x) =x 2 1, P 2 (x) =2x(x 2 1), P n+1 (x)p n 1 (x) =(P n (x)) 2 (x 2 1) 2, n =2, 3, S n P n (x) n,2 k n S n ( ) k n 3 (1999 ) {a 1,a 2,a 3, } (i) n, a n =1or 1, (ii) m, n, a mn = a m a n, (iii) n a n = a n+1 = a n+2 212 3 A = a b A 1 1 = d b c d ad bc c a A 1 4 (PSTP 141) a, b, c, d a 3 + b 3 + c 3 a + b + c + b3 + c 3 + d 3 b + c + d + c3 + d 3 + a 3 c + d + a + d3 + a 3 + b 3 d + a + b a 2 + b 2 + c 2 + d 2 x3 + y 3 + z 3 x + y + z x2 + y 2 + z 2 3 x + y + z =1

21 41 1 x, y, z, (x + y + z) 2 a(x 2 + y 2 + z 2 ), a 2 (1977 ) (m, n) : (1 + x n + x 2n + + x mn ) (1 + x + x 2 + + x m )

42 22 221, 1 a n, a 2002 a 1 =3, a n+1 = a n 1 a n, (n 1) 2 (PSTP 116, The Mathematics Student 1978 11 )2 7 2, 7, 1, 4, 7, 4, 2, 8,, 6 3 (1983 ) p,2 n n p n 1 (1989 IMO, 1990 KMO) n a 1,,a n, 8 7 1 + + 1 20 a 1 a n

22 43 2 ( 308 ) 2000 n n k n=1, x x k=1 3 (1979 ) n,3 n +1, 1 222 4 (2002 KAIST, Geometric Transformations I 9 ) O, A A m, m P m O Q A PQ m, m P Q A, P A P = Q, A O P = Q, A O, m O Q QA 5

44 (Wohascum County Problem Book 69 ),,,, ( ),? 6 (1998 Senior ) f x : (i) f(999 + x) =f(999 x) (ii) f(1998 + x) = f(1998 x) f : (a) x f( x) = f(x) (b) T, x f(x + T )=f(x) (i) f(1998 + x) =f(999 + (999 + x)) = f(999 (999 + x)) = f( x) f(1998 x) =f(999 + (999 x)) = f(999 (999 x)) = f(x) (ii) f( x) = f(x) (ii) (a), f(x + 3996) = f(1998 + (1998 + x)) = f(1998 (1998 + x)) = f( x) =f(x) (b) 1 (2002 KAIST, Geometric Transformations I 25 ) A, B X, AX BX X?

22 45 2 13,,? 3 10 37, 23 37 223, 7 (, ) 5 ( ) P P, 5 P 4, P P P 2 X, Y XY XY P BC, X, Y, B, C, I f x, y I 0 λ 1 f(λx +(1 λ)y) λf(x)+(1 λ)f(y), f I (convex; ) (concave; )

46 (Jensen ) I f, x 1,,x n I 0 λ 1,,λ n 1 (, λ 1 + + λ n =1) f(λ 1 x 1 + + λ n x n ) λ 1 f(x 1 )+ λ n f(x n ) 8 (1981 ) A, B, C, 2 sin 3A +sin3b +sin3c 3 3 2, 1 n 4 n n 2 (1979 ) a, b > 0, a, A 1,A 2,b, a, G 1,G 2,b A 1 A 2 G 1 G 2 224 9

22 47 10 (1994 KMO ) a 0 x 5 + a 1 x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 =0 a 5 > 0 14a 3 +9a 1 +35a 5 < 0, 3 3, 14 14,? 11 (?) n, n,,? 1 (1985 ) 6, 8, 10 2 (2002 KAIST ) n, n ( ) 3 (2002 ) 1 n, 1 n n

48 225, 12 13 (1917 Eötvös 3 ) A B k A B k ( ),, 1 4n +3 2 (2001 Putnam B6 ; ML 126 ) a a n n lim =0 n i =1, 2,,n 1 a n i + a n+i < 2a n n?

23 49 23 231, 1,?,,,, A, a,,,,, 2 ( ( ) 136 ) O A, B A O O M, N AMN B, M, N

50 O A, AMN AM AN BMN AB B C ( B C = B ), AM AN = AB AC, AC = AM AN AB AC AB, C AMN BMN 3 (Fermat-Euler ) a n a φ(n) 1 (mod n), φ(n) n n Fermat a p 1 1 (mod p) (p, a p ), {1, 2,,p 1} p p {a, 2a,,(p 1)a} p p, 1 2 (p 1) a 2a (p 1)a (mod p)

23 51 (p 1)! p, a p 1 1 n C = {c 1,c 2,,c φ(n) } C a ac = {ac 1,ac 2,,ac φ(n) } a n ac i ac j (mod n) c i c j (mod n), ac n, c 1 c 2 c φ(n) ac 1 ac 2 ac φ(n) (mod n) c 1 c 2 c φ(n) n, a φ(n) 1, 1 6,,,6? 2 (Problem-Solving Strategies 1 E7) a 1,a 2,,a n 1 1, S = a 1 a 2 a 3 a 4 + a 2 a 3 a 4 a 5 + + a n a 1 a 2 a 3 =0 n 4 3 (1999 4 11 3 ) 100 a 1,a 2,,a 100 a n, a m a 2 n n a 2 m a m a2 m m a 2 n a n a m m a n a n n a m

52, i =20, 40, 60, 80, 100 a i = 1 i 5 a i =1,? 232,, 4 (2000 ) n 2 f(f(n 1)) = f(n +1) f(n) f : N N? f(n +1) f(n) =f(f(n 1)) > 0, f(n +1)>f(n) f f(f(n)) >f(f(n 1)) f(n+2) f(n+1) >f(n+1) f(n), n>3 n f(n 1) n +1, f(f(n 1)) >f(n +1), f [x], ( ) 5 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18,, 1, 4, 9, 16, n n + n + 1 2 1 (1981 ) x, n, [nx] > [x] 1 + [2x] 2 + [3x] [nx] + + 3 n

23 53 [t] t ( : [π] =3, [ 2]=1) 2 (2001 Putnam B4 ; ML 126 ) S 1, 0, 1 f : S S f(x) = x 1 x f (1) (S) f (2) (S) f (n) (S) = φ (, f (n) f n ) 3 (1986 IMO) x i, s = x i > 0 x, y, z y<0,(x, y, z) (x + y, y, y + z) y<0?

54 24 241 1 7 4 8 1? (1) (2) 2,,,,, : : : : :,,,

24 55,,, +?,?,,?,,, 1 (1990 2 ) 1 n, a + b =2c

56 a, b, c n 2 (1999 KMO ) n n, 1 1 4 2,2 242, 3 A 1 A 2 A 15? 4 (1986,1989 ) 1 2 1 n k 2 1 n n k=1 1? 2 (1973 ) {X n }, {Y n } X 0 =1, X 1 =1, X n+1 = X n +2X n 1 (n =1, 2, 3,) Y 0 =1, Y 1 =7, Y n+1 =2Y n +3Y n 1 (n =1, 2, 3,) X : 1, 1, 3, 5, 11, 21, Y : 1, 7, 17, 55, 161, 487, 1

24 57 3 ( 216 ) n,6 n +1,n+2,n+3,n+4, n+5,n+6 A, B, A B

58 25, -?,?,,, 251 1 (1976 IMO 4 ) 1976 1 1976 5, 2 5=2+3< 2 3=6, 2+n<2 n (n 3), 5 (1) 4 2+2 4=2 2,4 2, 4 (1 ) 1 (2) 1 1 n<(1 + n)

25 59 2, (1 ) (2), 2 3,2 2+2+2=3+3, 2 2 2 < 3 3 3, 2 3, 2 2 (3) 1976 = 2 + 3 658, 2 3 658 2 (1999 IMO Selection Tests 1 ) x, y Z h(x + y)+ h(xy) = h(x)h(y)+1 h : Z Z x = y =0, h(0)+h(0) = h(0) 2 +1, (h(0) 2 1) 2 =0 h(0) = 1 (x, y) =(1, 1), h(0)+h( 1) = h(1)h( 1)+1, h( 1)(h(1) 1) = 0 h( 1) = 0 h(1) = 1 (1) h(1) = 1 : y =1 h(x +1)+h(x) =h(x)+1, h(x +1)=1 h(x) 1 (2) h( 1) = 0 : (x, y) =( 2, 1), h( 1) + h( 2) = h( 2)h(1) + 1, h( 2)(1 h(1)) = 1 h, h( 2) 1, (h( 2), h(1)) = (1, 0) or ( 1, 2) (2a) h( 2) = 1, h(1) = 0 : y =1, h(x +1)+h(x) =1 1 101010 h(x) 1+( 1)x 2

60 (2b) h( 2) = 1, h(1) = 2 : y =1, h(x +1)=h(x)+1, 1 h(x) x +1 (1), (2a), (2b) h 1+( 1) x h(x) =1,, x+1 2 1 (1989 KMO) f : N N (N ) (i), (ii), (iii) (i) f (ii) m, n N f(mn) =f(m)f(n) (iii) m = n m n = n m f(m) =n f(n) =m f(30) 2 a 1 a 2,a 3,, 12345, 123457, 1234571, 12345718,,9 252 3 (2002 ) n, n n, n,6 1, 2, 3, 6, 1=1, 2=2, 3=3, 4=1+3, 5=2+3, 6=6,6

25 61 a, b ab ab c ab c = qb + r, (0 q a, 0 r<b), q r a b q = a 1 + a 2 + + a m r = b 1 + b 2 + + b n c = qb + r =(a 1 + + a m )b +(b 1 + + b n ) = a 1 b + a m b + b 1 + + b n a i b( b) b j (<b) ab ab 4 (1990 APMO) n 6 n, n 1 1:2

62 (1) n (2) n (1) 0,(2) 1 2, (1) n (2) n 1, n =4, 5, 5 (2001 Putnam B1 ; ML 126 ) n n 1, 2,,n 2 (, n ) k (k 1)n +1, (k 1)n +2,, (k 1)n + n, 1 2 3 n 0 0 0 1 2 3 n n n n (a) 1 2 3 n (b) 2n 2n 2n 1 2 3 n (n 1)n (n 1)n (n 1)n

25 63 (a),(b),,, 1 (MathLetter 127 Proposal) 3 3 3 1, 2, 3,,27,,, 3? 2 (2002 ) a, b 2 k n 1,n 2,,n k (a) n 1 = a, n k = b, (b) n i n i+1 n i + n i+1 (1 i<k)

64 26 261,, 1 (1979 ),4, 4 0 2 180, 6 360 180 4 90, 0 180, 6 540, 360 540 1 ( 1972 ) R, ar 2 + br + c dr 2 + er + f 3 2 < R 3 2 a, b, c, d, e, f

26 65 2 (2003 KMO 2 ) n m, m m,, I,, m, n 6 2m <n (a) n, I m, n (b) I n, m =3 n I = n 262 (1) (2) (3) compact set 2 (Sylvester) n( 3), n (Paul Erdös) ( ), d, d d 0

66 A A d0 d1 B D H C l B, C, D, A H H D, B D AB d 1 d 0 d 0,, 3, ( ), c, c (c ) c, c, c, c c c c c c c c c c c c c c c c c c c c

26 67 infinite descent, 4 a 2 + b 2 + c 2 =2abc,, a, b, c,, a, b, c, 4 4 0 1, a, b, c a, b, c 2a 2,2b 2,2c 2, a 2 2 + b 2 2 + c 2 2 =2 2 a 2 b 2 c 2 a 2, b 2, c 2, 2a 3,2b 3,2c 3 a 2 3 + b 2 3 + c 2 3 =2 3 a 3 b 3 c 3? a, b, c 2 2 k, a =2 k a, b =2 k b, c =2 k c, a, b, c a 2 + b 2 + c 2 =2 k+1 a b c a, b, c 4, 4 0 1, a, b, c, a, b, c 1 (1976 ) a 2 + b 2 + c 2 = a 2 b 2

68 2 (PSTP 1117) c/d (d <100) k c = k 73 (k =1, 2, 3,,99) d 100, [x] x 3 (PSTP 1112) 2n, n, n n,? 4 (A Problem Seminar 56) n 1,n 2,n 3, n k+1 >n nk (k 1) 1, 2, 3, 4, 263,, 5 ( Eötvös 1915 2, ) EFGH ABC

26 69 1 ABC A, AB A A E H A' C A F B G A BC > ABC,, 2 ABC A EF, E F BC A BC E H A h' C F h B G E, EBC ABC, A, B, C, ABC = 1 2 EFGH, ABC 1 2 EFGH

70 6 ( 1999 ) 1 ABCDEF, M m AD, BE, CF M m [ 1] M M<3, ( ), M, M> 3 3 <M<3 M, [ 2] m m 2,, m m>1,, 1 <m 2 m,,,, M > 3 1 M,,

26 71 M M, AD BE CF, C E D M A B M F C D D' E BE CF, M M 2, BE = CF, BC = EF(= 1) BCF FEC (SSS), BCEF, ABCDEF AD A B M M F C D D' E ( 1) AD BE = CF = M, AD < M C, E AD M D AD, M, AD < M M 3, O,

72 O A B F C O D E,6 OAB, OBC, OCD, ODE, OEF, OFA, A = C = E, B = D = F ACE BDF 4 X =( AB, CD, EF) Y =( BC, DE, FA) ( 3), X Y BC AB FA EF CD DE AB + CD + EF = 0 AD AD = AB + BC + CD = BC +( EF) BC AB CD, BC +( EF) BC AB CD EF ABCDEF M min = 3 M> 3

26 73 compact set compact set, compact set 1 (1973 ) P, Q ABCD, PAQ < 60 2 (1983 ), ( ) 3 (1985 ) A, B, C, D, AB, AC, AD, BC, BD, CD 1

74

3 31 311 1 (1999 ) x>1 n>1 1+ x 1 nx < n x<1+ x 1 n x = y n (y>1) y n 1 ny n <y 1 < yn 1 n ny n y 1 y>1 y n 1 + + y +1<ny n <y n (y n 1 + + y +1) y n 1 + + y +1<y n + + y n + y n <y 2n 1 + + y n+1 + y n

76 2 (1927 Eötvös 1 ) a, b, c, d m = ad bc ax +by m (x, y), cx +dy m 1 ( 57 2) x 7 2x 5 +10x 2 1=0 1 2 (500 Mathematical Challenges) a, b, c, d, a 2 + b 2 =2 a 2 + c 2 =2 c 2 + d 2 =2 b 2 + d 2 =2 ac = bd ab = cd 3 (1979 ) n 3 3 1/4 312 3 (1910 Eötvös 2 ) a, b, c, d, u, ac, bc + ad, bd u bc ad u

31 77 bc ad u 2bc, 2ad u bc+ad, u bc ad u, WLOG(Without Loss Of Generality) 4 a, b, c a b + c + b c + a + c a + b < 2 a b c a b + c + b c + a + c a + b < 1+ b c + b + c c + b =2 1 (1998 Intermediate ) x, 1 y, 1 z (xyz 1) x, y, z,? 2 (1989 IMO ) n 3 t(n), n ( ) : t(3) = 1 : n n t(2n 1) t(2n) = +1 6 6 1 3 (2002 KMO) x 3 +2y 3 +4z 3 + 2003xyz =0

78 32, 1 (1990 1 ) A 3, B 4, B, A,,?,, B,,,,? B, A B 3 A 1 2 321 2

32 79 n n, n 1 k n, n k n k 11 5 11=4+3+2+1+1 4, 3, 2, 1, 1 4 3 2 1 1 11=5+3+2+1 5, n k k n,, 3 6 ( ) (1) A B C D A D,

80 B C, 6 3 6 3 =20, B C - 2(20 ) (2) - 5, i r i, 5 r i - r i (5 r i ), r i 2 3 6, - 6 r i (5 r i ) 6 6=36 i=1 (1) (2),2(20 ) 36, 2 13 1 P P,, v, e, f e 3v 6 2 (yaglom MCwES 119a ; Sperner s lemma) T,,,, T 1, 2, 3 1, 2, 3, T, 1, 2, 3 [ ]

32 81 322 A B B A, 4 (1967 Putnam )1 n n, 1 n,1,2,3,,,,, ( ),, k k = p e1 1 pe2 2 per r k (e 1 +1)(e 2 +1) (e r +1)

82 (e i +1), e i =2f i, k =(p f1 1 pf2 2 pf r r )2 5 3 3 3 1 1 1 27 6,,?,, 1 1 1 6 6,6 27 6 (2002 )(x, y) = (0, 0), f(x, y) = x2 xy + y 2 2x 2 + y 2 f(x, y) =(x 2 xy + y 2 )/(2x 2 + y 2 )=k k x 2 xy + y 2 =2kx 2 + ky 2, k y (1 k)y 2 xy +(1 2k)x 2 =0 k y 2, D = x 2 4(k 1)(2k 1)x 2 0

32 83 x =0 k =1, x = 0 1 4(k 1)(2k 1) 0, 8k 2 12k+3 0 3 3 4 k 3+ 3 4, 3 3 4, 7 ( 164 ) a, b, c, d 1, a + b + c abc < 2 1 0 <m<3 y =(m +1)x m 2 2 (2002 KAIST )1 1000,1 1000,,,, 1000? 3 (1980 2 ) a 1 <a 2 < <a n n, 3

84 4 (1980 ) [0, 1] a, b, c a b + c +1 + b c + a +1 + c +(1 a)(1 b)(1 c) 1 a + b +1 323 8 f(n) f(2n) =f(n), f(2n +1)=f(n)+1, f(1) = 1 f(n) =7 3 n 2 2n 2 n 0,2n +1 n 1 f(n) n 2 ( 1 ) a r a 1 a 0 2 f(2n) =f(n) f(a r a 1 0) = f(a r a 1 )+0, f(2n +1)=f(n)+1 f(a r a 1 1) = f(a r a 1 )+1 n = 10110 2 f(10110) = f(1011) + 0 = f(101) + 1 + 0 = =1+0+1+1+0, f(n) =7 1111111 (2),2 10111111 (2), 3 11011111 (2) = 223 9 (1989 IMO ) a n a 0 = 3 4, a 1+an n+1 = (n 0) 2

32 85?cosine cos θ 2 = 1+cosθ 2 a 0 = 3 4 =cosα, α n = α 2 n, a n =cosα n =cos α 2 n n =0, n 1+cosαn a n+1 = 2 =cos α n 2 =cosα n+1 n +1, 10 ( ) 2,3 1 (PSTP 311) x, y f (1) f(x, x) =x (2) f(x, y) =f(y, x) (3) f(x, y) =f(x, x + y), f 2 (1999 ),, A A (, )

86, P, Q P, Q,, ( ) 100 Z, Z 100 3 (1999 IMO Selection Test) x 1,x 2, y 1,y 2, : x 1 = y 1 = 3, n 1 x n+1 = x n + 1+x 2 n, y n+1 = 1+ 1+yn 2 n>1 2 <x n y n < 3 y n

33 87 33 331 1 α 0 <α<π sin θ +sin(θ + α) F (θ) = cos θ cos(θ + α), 0 θ π α, F, F θ F (θ) =F (0), sin θ +sin(θ + α) cos θ cos(θ + α) = sin α 1 cos α, 0 θ π α (1), [sin θ +sin(θ + α)](1 cos α) =sinα[cos θ cos(θ + α)], sin θ +sin(θ + α) sin θ cos α sin(θ + α)cosα =sinα cos θ sin α cos(θ + α), sin θ +sin(θ + α) (sin θ cos α +sinα cos θ) [sin(θ + α)cosα sin α cos(θ + α)] = 0, sin θ +sin(θ + α) sin(θ + α) sin(θ + α α) =0,(1) 0, 0 <α<π 1 cos α>0, 0 θ<θ+ α π cos θ cos(θ + α) > 0

88, 2 ABCD P AP = BP, BAP = ABP =15 PCD, PCD 15 ABCD PCD, AD = CD = PD AP D 30, DAP = PAD =75, BAP =15, BAP CDP, AB CD M, N, P BAP MN, x = BAP y = CDP y = f(x) f(15 ), f f 1 (60 )=15, f(15 )=60

33 89 P P AD = P DA =15 P P A = PA, P AP =60 AP P, P P = P A = P D, PP D = 150 P PD P AD, PD = AD = CD, PC = CD PCD 1,?? 2 3, 332 3 ( ) 4 (1923 Eötvös 1 ) O r A, B, C A, B, C r

90?,,,,, 1 (PSTP 142, School Science and Mathematics 1981 2 ) AB C AB, AC, CB k 1, k 2, k 3 k 2 k 3 k 2, k 3 D, E C AB k 1 F FDCE 2 (1999 KMO ) ABCD, BD < AC E F AB CD, BC AD, L M AC BD LM EF = 1 AC 2 BD BD AC

34 91 34,,,, 341 1 3?, [ ],?

92,? 1, 3:0 2:1 (1) (2) m l l (1) 3:0 (2) 2 : 1 2:1,,2 m,(2) 2 1 α, α 4:0,3:1,2:2 (1) 4 : 0 : 0

34 93 (2) 3 : 1 : 4, α 3 β 4 (3) 2 : 2 : 3, α A B, C D α A, B AB, CD A, B, C, D, A CD C D AB, A AB CD β β C α 3,(2) (3) 7? 2 (1999 1 ) a 1 <a 2 < <a n a 1 a 4 2 + a 2 a 4 3 + + a n a 4 1 a 2 a 4 1 + a 3 a 4 2 + + a 1 a 4 n n =3, 1 (1999 ) 2 1999 k=0 4sin 2 kπ 2 2000 3 2 (1999 KMO ) ABC DEF AD, BE, CF AA = a, BB = b, CC = c A, B, C A BC, B CA, C AB M, M ABC d, 1 a + 1 b + 1 c = 1 d

94 3 (1998 KMO ) 1, 2, 3,, 2000 ( ) 2000 342 3 (1974 ) a, b, c a a b b c c (abc) a+b+c 3,,, [ ] a, b a a b b (ab) a+b 2 ( ), 1 a, b ( ) a b a 2a b 2b a a+b b a+b a a b b a b a a b 1 b a b, 1, a b 0

34 95 2 a a b b (ab) a+b 2 b b c c (bc) b+c 2 c c a a (ca) c+a 2 (a a b b c c ) 2 a 2a+b+c 2 b a+2b+c 2 c a+b+2c 2 (a a b b c c ) 2 (abc) a+b+c 2 a a 2 b b 2 c c 2 (a a b b c c ) 3 2 (abc) a+b+c 2 2 3, ( ) a a b b a b b a 1 (+) ( ) 4, 4 4 4,

96 ( ),,, ( ), ( ) ( ) 4 4 0, 1, 2, 3, 4 2 ( ; MathLetter 133 ) n (a i ), (b i ),, (z i ) 0 a 1 b 1 z 1 + a 2 b 2 z 2 + + a n b n z n a 1b 1 z 1 + a 2b 2 z 2 + + a nb n z n, (a i ), (b i ),, (z i ) (a i), (b i ),, (z i )

35 97 35,, 351 1 n k 2 2 k k=0 (1/2) k x k, f(x) = k 2 x k n k=0 x k = 1 xn+1 1 x (x = 1) x n k=1 kx k 1 = 1 (n +1)xn + nx n+1 (1 x) 2 k x, n k=1 n k=1 kx k = x (n +1)xn+1 + nx n+2 (1 x) 2 k 2 x k 1 = 1+x (n +1)2 x n +(2n 2 +2n 1)x n+1 n 2 x n+2 (1 x) 3 n k=1 k 2 x k = x + x2 (n +1) 2 x n+1 +(2n 2 +2n 1)x n+2 n 2 x n+3 (1 x) 3, f 1 = 2 n k=0 k 2 2 k =6 n2 +4n +6 2 n

98, 2 (Crux Mathematicorum 1979 1 ) a, b, c 1, 4(1 + abc) (1 + a)(1 + b)(1 + c), 2 n 1 (1 + a 1 a 2 a n ) (1 + a 1 )(1 + a 2 ) (1 + a n ) ( ), n ( ) n =1 n =2 2(1 + ab) (1 + a)(1 + b) 1+ab a + b (a 1)(b 1) 0 n ( ), n +1 2 n (1 + a 1 a n+1 )=2 2 n 1 (1 + a 1 a n 1 (a n a n+1 )) 2 (1 + a 1 ) (1 + a n 1 )(1 + a n a n+1 ) ( n ) (1 + a 1 ) (1 + a n 1 )(1 + a n )(1 + a n+1 ) ( 2 ) 3 (PSTP 119, American Mathematical Monthly 1946 2 )(x + y) 1000

35 99 1 n n 1 k k 2 k=0 k 2 (1999 ) 99 C 1,C 2,,C 99, n S n :( P ) C i C j (1 i, j 99), C i C j S, C j C i S 3 (2002 ) n a 1,,a n x 1,,x n ( ) a 1 a 2 a n ( ) x k x n (k =1,,n 1), (a 1 + x 1 a 2 )(a 2 + x 2 a 3 ) (a n 1 + x n 1 a n )(a n + x n a 1 ) a 1 a 2 a n (x 1 +1)(x 2 +1) (x n +1) 352, -, :) A, B, C P, A, B, D, E Q, E, D C, Q P A, B, C P,

100 4 (1913 Eötvös 1 ) n>2, (1 2 3 n) 2 >n n n 2, n, 2n, n? (1 + k)(n k) n ( ) (0 k<n) (1 + k)(n k) n = k(n (k +1)) 0 ( ), k =0,n 1, (1 2 3 n) 2 = (1 n)(2 (n 1)) (n 1) n n n = n n, n>2 0 <k<n 1 5 (1926 Eötvös 1 ) a b, x, y, s, t x + y +2s +2t = a 2x 2y + s t = b x + y +2s +2t =1 x + y +2s +2t =0 2x 2y + s t =0 2x 2y + s t =1

35 101 (x 1,y 1,s 1,t 1 ), (x 2,y 2,s 2,t 2 ), a, b (x, y, s, t) =a(x 1,y 1,s 1,t 1 )+b(x 2,y 2,s 2,t 2 ) (x, y, s, t) =(1, 0, 1, 1), ( 1, 1, 1, 0) (a, b) =(1, 0), (0, 1) (x, y, s, t) =a(1, 0, 1, 1) + b( 1, 1, 1, 0), 6 (2002 IMO 1 ; ML 128 ) n T x + y<n (x, y) T (x, y), x x, y y T (x,y ) X- x- n, Y - y- n X- Y - A X-Y - X(A), Y (A), X(A), Y (A) X(T ) = Y (T ) x m := (x = m ) y m := (y = m ) (0 m n) X(T ) x m, Y (T ) y m x m ( ) X n y m Y n ( )

102 X(T ) = n x m = m=0 n y m = Y (T ) m=0 (1) n =1 : T,, X 1 = Y 1 = {0}, X 1 = Y 1 = {1} (2) n k n ( ) n = k +1 x + y = k : (p, q) p + q = k A x p, y q (x, y), B C B C T, X n = X p (B) {0} X q (C) Y n = Y q (C) {0} Y p (B) p, q k X p (B) =Y p (B), X q (C) =Y q (C), X n Y n x + y = k : A k+1 A k 1 1, B k+1 B k 1 1 A k = B k A k+1 = B k+1, n = k +1,( ), (1) (2) ( ),

35 103 1 (a, b, c, d) a>b>c>d>1, a! d! =b! c!, a, b, c, d 2 (1999 KMO ) N, f : N N?, (1) f(2000) = 1999, (2) n, f(n),f(2n),f(3n),,f(kn),, (3) f(f(n)) = 1 3 ABCD [ ] (1) A B C D ABCD 1/9 (2)

104

4,,

106?,, 1 A B A B A B? 2 n a 1,a 2,,a n f(x) =a 1 sin x + a 2 sin 2x + + a n sin nx

107 x f(x) sin x a 1 +2a 2 + + na n 1 3 (1990 KMO 2 ) (1) i =1, 2, 3, P i = x i(i 1) <x 2 i(i +1),x 2, i = j P i P j = ( ) (2), f f(m, n) = 1 (m + n 2)(m + n 1) + m 2 f(m, n) =f(p, q) m = p, n = q 4 (1990 Putnam) T n T 0 =2, T 1 =3, T 2 =6, T n =(n +4)T n 1 4nT n 2 +(4n 8)T n 3 (for n 3) 2, 3, 6, 14, 40, 152, 784, 5168, 40576, 363392 {A n } {B n } T n = A n + B n T n, T n

108

5 1 (Yaglom MCwES 9 ) 11,, 2 A, B, C 25 A B C 2, A A A, C, B? 3 (A Problem Seminar 57) 4 (1903 Eötvös 3 ) A, B, C, D, k 1 B, C, D, k 2 A, C, D, k 3 A, B, D, k 4 A, B, C B k 1 k 3 C k 2 k 4

110 5 (1915 Eötvös 1 ) A, B, C n>ν ν An 2 + Bn + C<n! 6 (1974 ) 1 2 7 (1977 )0<p<q, a, b, c, d, e [p, q], 1 (a + b + c + d + e) a + 1 b + 1 c + 1 d + 1 2 p q 25 + 6 e q p, 8 (1986 ) K 1, K 2 A B AB K 1 K 2 K 1 P T- (, ), CD AB CPD K 1 C, D 9 (1988 IMO) f : n, f(1) = 1,f(3) = 3, f(2n) =f(n), f(4n +1)=2f(2n +1) f(n) f(4n +3)=3f(2n +1) 2f(n) f(n) =n 1988 n 10 (1990 IMO) n n 3 2n 1 E k,, E n k k k

111 11 (1990 IMO) Q + x, y f(xf(y)) = f(x) y f : Q + Q + 12 (1990 KMO ) A, B (1) 1 A, (2) A 2 k +2(k =0, 1, 2,), (3) B 2 k +2(k =0, 1, 2,), 1988, 1989, 1990 A B 13 (1999 ) n>1, a 1,a 2,,a n d (i) a k a k 1 = d for k =2, 3,,n, (ii) a i = b m i i for i =1, 2,,n m i (> 1) b i 14 (1999 KMO )3 a,(a 2 1)(b 2 1) = c 2 1 2 (b, c) 15 (1999 KMO ) f f(x) : x, f(x +1)=f(x)+1, f(x 3 )=(f(x)) 3 16 (2002 IMO 3 ; ML 128 ) a m + a 1 a n + a 2 1 a m, n 3 17 (2002 IMO 6 ; ML 128 ) n 3,Γ 1, Γ 2,,Γ n 1 O 1,O 2,,O n

112 1 i<j n 1 (n 1)π O i O j 4 18 (MathLetter 129 Proposal) n n, n n 19 (2002 APMO) x i? n x1 + + x n x 1! x 2! x n!! n (, [y] y ) 20 (2002 APMO) a, b, c 1 a + 1 b + 1 c =1 a + bc + b + ca + c + ab abc + a + b + c 21 (2002 ) N = {0, 1, 2,} x, y N xf(y)+yf(x) =(x + y)f(x 2 + y 2 ) f : N N