2 KAIST 1988,,KAIST MathLetter, 3,,, 3,, 3, 3,

(M 2 )

2 KAIST 1988,,KAIST MathLetter, 3,,, 3,, 3, 3,

3,,, 2003 8,

4

1 7 11 8 12 26 2 39 21 40 22 54 23 67 24 80 3 93 31 n! 94 32 101 33 115 4 131 41 132

6 42 146 5 163 51 164 52 180

1

8 11 4 4?!,? 2??,? (Dirichlet),, 2 :(n +1) n 2

11 9? 1 n (n +1), 2 1 1 ( ), (,, ), (,, ), (,, ), (,, ) (,, ) (,, ),(,, ) (,, ) 2 ( ),,,,,,

10 3 ( ) (2 ) (3 ),? ( ),, n, (n +1) 2, 1 20 11, 1 20 10 ( ) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 11 ( ),,

11 11,?,,,? 1 k, k +1, kn +1, n, k +1, q 1 = q 1 = = q n = k 2 q 1,q 2, n (q 1 + q 2 + + q n )+1 1 n n i (1 i n), i (q i +1) i (1 i n) i q i, (q 1 + q 2 + + q n ), i q i +1

12?,? 3 14 3 5 1 3 4 3 4=12 14,5 2,14 (> 4 3) 3 5 (= 4+1) 4 67 i (i =1,,12) i i 1 i i 1 2 1, 3 2,, 12 11

11 13 0+1+2+ +11=66 67, 2 67=(0+1+2+ + 11) + 1 q 1 =0, q 2 =1,, q 12 =11 (q i = i 1) 2 i q i +1(= i) n m 1, m 2,,m n m 1 + m 2 + + m n n r ( / / ), m i r ( / / ) i m i r, m 1 + m 2 + + m n n n r + r + + r = nr n n r, 5 2 5 2

14 1 4 5 2 2

11 15 111 10 101 2 10 100 101 2 1 2, 2 3 55 1

16 112 1 2n (n +1) 1 2n, 1, 3, 5, 7, n 1 2n 2n n (1, 2), (3, 4), (5, 6),, (2n 1, 2n) n +1,, 2, 4, 6, 8,, 2n n 2, n +1, 2, 4, 6,,2n n,

11 17, n +1 n? n,? 12 0 20,

18 113 n ( ), (, A B B A ) n, (0) (n 1 ) n, 0, 1, 2, 3,, n 1 (n 1), n, n

11 19 114 n a 1,a 2,,a n n n n 1 n n 1 1 2 n 1,?,? 2 n (n +1), n n? n,, (a 1 + a 3 ) (a 2 + a 3 )=a 1 a 2 (a 1 ) (a 2 ),?,! n a 1, a 1 + a 2, a 1 + a 2 + a 3,, a 1 + a 2 + + a n n a 1 n, n n, n n, n 0, 1, 2,, n 1 n 1, n

20 (1) n, a 1 + + a i a 1 + + a j (i<j) (a 1 + + a j ) (a 1 + + a i )= a i+1 + + a j (2) a 1,a 2,,a n ( ), n, a 1,a 2,,a n n n, n r 1,r 2,,r n 1 n

11 21 115 8, 4,, 4 1/8 8, 4, 8 4=32, 8 4 ( 3 8 3=24, 32 ) (1989 KMO) 10 10, 2

22 1 : 111 kn +1, n, k +1 (1) 100 15 112 (2) 100 4 (3) 100 9 4 33, 113 1/2 114 141 2 1 5

11 23 (1906 Eötvös, 1988 KMO) n, 115 a 1,a 2,,a n 1 n, (a 1 1)(a 2 2) (a n n) (1925 Eötvös ) 4 a, b, c, d 116, (a b)(a c)(a d)(b c)(b d)(c d) 12 17 20, 117 5 2 5, 118 1

24 (1988 KMO) 4 9 119 3 2, x, y 1110 5, ( ) 1, 4, 7,,100 20 A 1111 A 104 52, 100 1112

11 25 111 6, 6 3,, 1??, A B B A (Ramsey),6 112 3 2,3 12 113 12 (84 ) 23

26 12 (Carl Friedrich Gauss)? 1 100 1 100 101 2 99 101, 50 51 50 101 (1 + 100) + (2 + 99) + + (50 + 51) = 50 101,?

12 27 1,?,, n a n a 1, a 1 =1, a 2 2? a 3 3 a 4 4? 3 a 3 =3, 2 a 2 =2?, 4 a 4 = a 3 + a 2 =3+2=5 a 3 a 2 a 4 = +!, 5

28 a 5 a 4 a 3, a 5 = a 4 + a 3 =5+3=8, n, (n 1) (n 2) a n = a n 1 + a n 2 a n 1 a n 2 a n a 1 a 2 a n-1 a n-2 a n = + a 1,a 2,a 3, 1 2, 1, 2, 3, 5, 8, 13, 21, 34,,,, (recurrence) 2n

12 29 2n 2 1 z n z 1 1, z 2 2 2 2 z 3 =3, z 4, z 5? z 4,, 2 3 ( ), z 3,

30,,, z 4 = z 3 +1=3+1=4 z 5, z n 2 (n 1) z n 1,, z 1 =1, z n = z n 1 +1 z n, n 1 z n 1 z n = n,

12 31 121 2 3, a b (a = b ) ab + a + b (1) 109 (2) 143 (3) 191 (4) 257 (5) 323 ab + a + b 2, 3, 11, 35, 47, 107,,? c c = ab + a + b, 1 c +1=(a +1)(b +1), 1 c = a b (1 )3,4,, 3 m 4 n ( 1 ) 1 110, 144, 192, 258, 324, 3 m 4 n 110 258 (1), (4) 7 7, 7 2001

32 122 ( ) 10% 1 n?,, 1 n a n a n 1 a n 1 10% 010a n 1, a n = a n 1 +010a n 1 =110a n 1, a 0 =1, a n =110a n 1 1, 1 1 1, 10?(110 10 26 )

12 33 123 0 1 bit string, 0110100 8 bit string n 0 bit string a n, n 0 bit string, 0 1 n 3 ( ) n 1 0 bit string n 1 0 bit string 1, a n 1 0, n 0 bit string 0, 1, n,0, 0 bit string n 2 0 bit string 10, a n 2, n 3 a n = a n 1 + a n 2 a 1 =2,a 2 =3 3 0 12 bit string

34 10 121, 10? ( ) 10% 1, 100 122 1 1,10?, 110 10 26 1, 2 123 10,10? n a n 124 a 1 =1, a n = n +1 n 1 (a 1 + a 2 + + a n 1 ), a 2000 (n>1) 1, 2, 3,,n, 125 2 ( )

12 35 126 n h(n) =1+ 1 2 + 1 3 + + 1 n, h(1) = 1, h(2) = 1+ 1 2, h(3) = 1+ 1 2 + 1 3 n =2, 3, 4, n + h(1) + h(2) + h(3) + + h(n 1) = nh(n) n, 127? 1 1 2 1 2 4 3 5 3 1 4 2 6 7,, 128 10 ( ) 129, 10?

36 1210 1 2, 1 3, 1 10 10? 2 1, 2, 3 10 1 2 1211 (, 1 1 3,2 2 3 ) 10?, 1212 1 2 n 2 n 1 1 2 3 n 1 n 2 3 4 n b n, b n+2 = b n+1 + b n 2 n 2 1 1213

12 37 3 n 121? 3 2n 2 1 3n 122?

38

2

40 21 3x +5y =7?, 119x + 271y =1?, x = 1 271y 119,1 271y 119 y 119k, 119k +1,,119k + 118 119, 3974x + 2771y =1?,? a, b, gcd(a, b) =ax + by x, y x, y,,

21 41 x, y 1 7497x + 19278y = 1071 x, y gcd(7497, 19278) 2 7497 19278 14994 7497 4284 1 4284 1 3213 4284 3213 3213 1071 3 3213 0 4284 = 3213 1 + 1071 1071 = 4284 3213 7497 = 4284 1 + 3213 3213 = 7497 4284 19278 = 7497 2 + 4284 4284 = 19278 7497 2 1071, 3213, 4284, 1071 = 4284 3213 = 4284 (7497 4284) = 7497 + 4284 2 = 7497 + (19278 7497 2) 2 = 19278 2 7497 5 1071 7497 19278,(x, y) =( 5, 2)

42 x, y 7497 = 1071 7, 19278 = 1071 18 lcm(7497, 19278) = 7497 18 = 19278 7, x, y x = 5+18k, y =2 7k (k ) 2 52 = 1482x + 1274y x, y 1482 1274 1 1274 6 208 1274 1248 208 26 8 208 0 2 0 26 52 = 2 26 = 2 (1274 6 208) = 2 1274 12 208 = 2 1274 12 (1482 1274) = 14 1274 12 1482, (x, y) =( 12, 14) 3 a, b

21 43 (1) 123a + 321b =21 (2) 770a + 1015b =35 (3) 630a + 1386b = 126 (4) 1835a + 2337b = 246 (5) 2465a + 3132b = 319? 4 3x 2(mod5) 1 ( ) 3x 3 (mod 5) x 1 (mod 5) 5, 3 2 ( ) 6x 4 (mod 5) x 4 (mod 5) 2,

44 3 ( ) 5 3x 2,3x 2=5k, 3x 5k =2 (k x ) (x, k) =( 1, 1), (x, k) =( 1, 1) + (3, 5)t x, x = 1+3t, x 1(mod5) [ 1] [ 2] (1) ca cb (mod m) a b (mod m) (, (c, m) =1, ) (2) da db (mod dm) a b (mod m) (3) ea eb (mod m) a b (mod m/d) (, d =(e, m)) (3) (1) (2),(1) (2) (3) d e/d [ 1] [ 2] [ 3],? 119x 15 (mod 23), 15 119 [ 1- ], 119x (mod 23) x [ 2- ], [ 3] [ 1] 2 5 [ 3] k = 1 5k, [ 1] 5 119x 15 (mod 23)

21 45 119x 23k =15 119 23 5 115 5 4 23 20 4 3 1 3 1 1 = 4 3 = (119 115) (23 20) = (119 5 23) (23 5 4) = 119 6 23 + 5(119 115) = 6 119 31 23, 15 = 15(6 119 31 23) = 90 119 465 23,(x, k) =(90, 465) k k 11 (mod 119) 11 23 = 253 ( ), 119x 238 (mod 23) (119, 23) = 1, 119 x 2 (mod 23) [ ] [ ],, [ ],,

46 [ 2] [ 1] 3x 2(mod5) 3x (mod 5) x, 3u 1 (mod 5) u u 3ux 2u (mod 5), x 2u (mod 5), (3x 2) (3u 1),, 1 3 2 1(mod5),2 (mod 5) 3, 3 2, ( 1 ) 2 3 1,(mod5) 2 1 3, 3 1 2 3 3 6 119x 15 (mod 23) 119 119u 1(mod23) u 119 1 =6 119 31 23, 119 6 1, 119 =6 119 =6, x 90 2 (mod 23)

21 47 (modm) a a, au 1(modm), a m,4u 1(mod6)?

48 211 x + y 3 (mod 4) x y 1 (mod 4), x, y 2x 4 (mod 4), 2y 2 (mod 4), x, y,(x, y) =(0, 1)? 2x 4(mod4), x x 0 (mod 4) y 3(mod4), x 2(mod4) y 1 (mod 4),(x, y) (0, 3) (2, 1) (mod 4) x +2y 0 (mod 3) 2x + y 0 (mod 3)

21 49 212 x +2y +3z =3 x, y, z x, y, z z m x +2y =3 3m (x 0,y 0 )=( (3 3m), 3 3m), (x, y) =( (3 3m), 3 3m)+k( 2, 1), m, k, x = 3+3m 2k, y =3 3m + k, z = m 3x +4y +6z =5 x, y, z

50 211 (1) 7x 3 (mod 15) (2) 12x 6 (mod 15) (3) 5x 25 (mod 35) 212 3x +5y 1 (mod 12) 2x 3y 3 (mod 12) (a, b) 213 (1) 1845a + 984b = 123 (2) 1751a + 1377b =51 527x + 3193y =403 x, y 214, 215 (1) ca cb (mod m) a b (mod m) (, (c, m) =1) (2) da db (mod dm) a b (mod m) (3) ea eb (mod m) a b (mod m/d) (d =(e, m))

21 51 216 (1) 12378x 6 (mod 3054) (2) 172x 1000 (mod 20) 217 x +2y +3z = 4 2x z = 1 218 3x +6y + z = 2 4x +10y +2z = 3 219, ax 1(modm) (a, m) =1

52 (mod m) 2110, a b m a b (mod m) a b (mod m) a, b, c, d, 2111 ax + by + cz = d gcd(a, b, c) d x +2y +5z +9w =5 x, y, z, w 2112, 1100 2113 700 65700?

21 53 3a +4b +9c +12d =4 211 a, b, c xbc + yca + zab = n 212

54 22?,?,,?,???? n? 2, 3, 5, 7, (, ) 1

22 55,,?, 30 40 2, 3, 5 1 2 3 5+1 = 31 1, 61, 91, 121, 31 30 1 2, 3, 5 1, x 1 (mod 2) x 1 (mod 3) x 1 (mod 5) x 1 (mod 30)? 2, 1, 2 30 40?

56, 4k +1 5, k 5l, 5l +1, 5l +2, 5l +3, 5l +4 4k +1 k : 5l 5l +1 5l +2 5l +3 5l +4 4k +1 : 20l +1 20l +5 20l +9 20l +13 20l +17 5 2 20l +17, 17, 37, 57, 77, 97, 30 40, 37?, (, algorithm), m 1,m 2,,m n a 1,a 2,,a n m 1 m 2 m n, x a 1 (mod m 1 ) x a 2 (mod m 2 ) x a n (mod m n ) (m 1,m 2,,m n ) x, mod m 1 m 2 m n

22 57, m 1,m 2,,m n? 2, 3, 5 4, 2, 3, 5, 7 (?)? 4 6 3 4 3,6 4 x 4 3,6 4, x 4 6, 4 4 2,6 4 4 2,6 4 x 2k x 4 2 k 2 1, x 6 4 k 3 2, x =2k, 2k 2 (mod 4) k 1 (mod 2) 2k 4 (mod 6) k 2 (mod 3)

58 2 3, k 5 (mod 6) x k =5 10 m 1,m 2,,m n, x ( 3), m 1,m 2,,m n ( 4), 5 x 1(mod2),x 0(mod3),x 3(mod5) x 2, 3, 5 p, q, r? p 2 1,3 5 q 3 1,2 5 r 5 1,2 3 p, q, r, x x p 1+q 0+r 3 (mod 2 3 5)

22 59 1, 0, 3, x = p 1+q 0+r 3+2 3 5 k 1 1+ 0 0+ 0 3+0 k =1 0 1+ 1 0+ 0 3+0 k =0 0 1+ 0 0+ 1 3+0 k =3 (mod2) (mod3) (mod5), p, q, r p 3 5 p =15p,15p 1(mod2) p p =1, p =15 q, r,10q 1(mod3) q =1,q = 10, 6r 1(mod5) r =1,r =6 x 15 1+10 0+6 3=33 3 (mod 2 3 5 = 30) (q 0 ) ( ) m 1,m 2,,m n, x a 1 (mod m 1 ) x a 2 (mod m 2 ) x a n (mod m n ), (mod m 1 m 2 m n ) k =1, 2,,n M k = m 1m 2 m n m k

60, M k m i m k m 1,,m n, M k m k,, A k M k 1 (mod m k ) A k P k = A k M k, P k 0 (mod m i ), i = k, P k 1 (mod m i ), i = k x = P 1 a 1 + P 2 a 2 + + P n a n k x 0a 1 +0a 2 + +1a k + +0a n = a k (mod m k ), x, x,modm 1 m 2 m n y x a k y (mod m k ), m k x y m 1,m 2,,m n x y, m 1 m 2 m k x y x y (mod m 1 m 2 m k ), x,

22 61 221 100,,,,,!? x 4 (mod 5) x 3 (mod 4) x 2 (mod 3) ( 100 <x<150 ) 1 ( ) x 5, 4, 3 4, 3, 2,, x 1 (mod 5) x 1 (mod 4) x 1 (mod 3), x 1(mod60=5 4 3), x = 119 2 ( ) x =5k +4, k 4 k 4l 2, 4l 1, 4l, 4l +1, x

62 20l 6, 20l 1, 20l +4, 20l +9, 4r +3 20l 1 l 3m, 3m +1, 3m +2, x 60m 1, 60m + 19, 60m +39, 3r +2 60m 1 3 ( ) x =5k +4 5k +4 3(mod4), k 3(mod4) k =4l +3 x =20l +19 20l +19 2(mod3) l +1 2(mod3) l 1(mod3), l =3m 1 x =60m 1 4 (!) P 1 =4 3 A 1 1(mod5) A 1 =3,P 1 = 36 P 2 =5 3 A 2 1(mod4) A 2 = 1, P 2 = 15 P 3 =5 4 A 3 1(mod3) A 3 = 1, P 3 = 20, x 4P 1 +3P 2 +2P 3 =59 1 (mod 60),, 1, 2 100,?

22 63 222 x 2(mod6),x 5(mod7),x 5(mod9)x 3 (mod 11) 6, 7, 9, 11 x 2(mod6) x 2 0(mod2) x 2(mod3) x 2 (mod 3) x 5(mod9), x 0 (2), x 5 (7), x 5 (9), x 3 (11), m 1 =2,m 2 = 7, m 3 =9,m 4 = 11; M 1 = 693, M 2 = 198, M 3 = 154, M 4 = 126; a 1 =0, a 2 =5,a 3 =5,a 4 =3; M = 1386 x a 1 P 1 + a 2 P 2 + a 3 P 3 + a 4 P 4 (mod M) P 1, P 2, P 3, P 4 a 1 =0 P 1 P 2 = M 2 A 2 = 198A 2 1 (mod 7) P 3 = M 3 A 3 = 154A 3 1 (mod 9) P 4 = M 4 A 4 =126A 4 1 (mod 11) ( ), A 2 =4,A 3 =1,A 4 = 2, P 2 = 792, P 3 = 154, P 4 = 252, x a 1 P 1 + a 2 P 2 + a 3 P 3 + a 4 P 4 = 3960 + 770 756 184 (mod 1386) x 1(mod3),x 2(mod4),x 3(mod5)x 4 (mod 6)

64 223 k, k n +1,n + 2,, n + k ( ), k p 1, p 2,,p k, n+1 p 1, n +2 p 2,,n + k p k n +1 0 (mod p 1 ) n +2 0 (mod p 2 ) n + k 0 (mod p k ), mod p 1 p 2 p k, n,, k ( ) n =(k +1)!+1 n +1=(k +1)!+2 2, n +2=(k +1)!+3 3,,n + k =(k +1)!+(k +1) (k +1) k, k a 1,a 2,,a k i (1 i k) (p i ) i a i p i

22 65 221 2x 1 (mod 5) 3x 3 (mod 7) 4x 5 (mod 9) 4000 222 17 3 16 10 15? 3 223 2 4 2 5 2 60? x 2 1(mod56) 224

66 (p 1)!+1=p k p k ( 221 ) 2 ( ) 222, 2,,?, 3,,?

23 67 23 1 36 36 +41 41 77 41 36 (mod 77) 36 36 +41 41 36 36 +( 36) 41 =36 36 (1 36 5 ) (mod 77) 36 5 1(mod7) 36 5 3 5 = 243 1 (mod 11) 36 5 (mod 77), 2 (1894 Eötvös ) 2x +3y 9x +5y 17 17 2x +3y 17 4(2x +3y) 17 8x 12y + 17(x + y) 17 9x +5y 17 3 (1976 ) a 2 + b 2 + c 2 = a 2 b 2

68 4 0, 1 a, b mod 4 2 3 1 a, b 4 4 mod 4 a, b, c a, b, c 0, 2, a =2 m a 0, b =2 m b 0, c =2 m c 0 a 0, b 0, c 0 m 1 a 2 0 + b 2 0 + c 2 0 =2 2m a 2 0b 2 0, 4 4 mod 4 a 0, b 0, c 0, (a, b, c) =(0, 0, 0) 4 a, b, c ab, bc, ca a, b, c (abc) 2 = ab bc ca = m 3 n 3 r 3 =(mnr) 3 (abc) 2, ( ), abc, a = abc/bc, b = abc/ca, c = abc/ab a 2 + b 2 = c 2

23 69 (a, b, c) (3, 4, 5), (5, 12, 13), (6, 8, 10) (a, b), (a, b, c) 5? 1,, 1 gcd(a, b, c) =1 ( ) 2 gcd(a, b) =gcd(b, c) =gcd(c, a) =1 3 c, a b,, b

70 4 b 2 = c 2 a 2 =(c + a)(c a), ( b 2 )2 =( c+a 2 c a )( 2 ) 5 gcd( c+a 2, c a 2 )=1 6 (b 2 ), m, n 2 = m 2, c a 2 = n 2 7 (a, b, c) =(m 2 n 2, 2mn, m 2 + n 2 ) c+a, m n m >n, (, m, n, r ) (m 2 n 2 )r, 2mnr, (m 2 + n 2 )r 1 (a, b, c) gcd(a, b, c) = d > 1, ( a d, b d, c d ) ( a d )2 +( b c )2 =( c d )2, (a, b, c) ( a d, b d, c d ), 2 3 2 gcd(a, b) =1 a, b a, b, c, 2 0(mod4) ( mod 4 1, 0) a, b, c

23 71 4 5 6 7 6, 1, r (m 2 n 2 )r, 2mnr, (m 2 + n 2 )r (Fermat) Fermat n 3, x n + y n = z n,, 1453 (arithemetica) 1621 3, x 2 + y 2 = z 2

72 x, y, z n 2 x n + y n = z n ( 0 ) 1630,,, 350 17, n =3: (Euler) n =5: (Dirichlet) n =7: (Legendre) 1930 617 30,000, 1976 125,000, (A Wiles) 1994,, 1994 10 Wiles Taylor 1995 Annals of Mathematics 5

23 73? n =4,n =5,, (Conjectures),, x n + y n + z n = c n n 4 Noam Elkies, 2682440 4 + 15365639 4 + 18796760 4 = 20615673 4,

74 231 2x 2 6x +(1+a) =0 a b, c 2(x b)(x c) =2x 2 2(b + c)x +2bc =0 b + c =3, 2bc =1+a a 1+a 2, bc 1, b c b + c =3 {b, c} = {1, 2}, a =2bc 1=3 x 2 +(m +1)x +2m 1=0 m

23 75 232 a 2 + b 2 + c 2 =2abc a, b, c a, b, c 4 4 1, 0 mod 4, 4 a, b, c 2, a =2 m A, b =2 m B, c =2 m C A, B, C m 0 a, b, c m 1 2 2m A 2 + B 2 + C 2 =2 m+1 ABC 4 4 mod 4 A, B, C,, a 2 + b 2 + c 2 + d 2 =2abcd

76 231 17p +1 p 6 232 6 a + b + c 6 a 3 + b 3 + c 3 233, x 234? 1 3 5 7 x 11 (2) 1000 (2) 1101 (2) 10010 (2) 110101 (2) N 7 235 3 N?

23 77 m 236, m + n +1 mn +1 n (1899 Eötvös ) n 237 1897 A = 2903 n 803 n 464 n + 261 n (1999 ) 238 1 x, 1 y, 1 z (xyz 1) x, y, z,? (2002 KMO) x 3 +2y 3 +4z 3 +8xyz =0 239 (2002 KMO) a 2310 (x 1,x 2, 2002 ) (1) x 1 x 2 x 2002 0 (2) 0 <x 1 + x 2 + + x 2002 a (3) x 2 1 + x2 2 + + x2 2002 +32 = a 2

78 2311 a 2 + b 2 = c 2,gcd(a, b, c) =1 a, b, c a, b gcd(a, b) =1 gcd( a+b 2312 2, a b 2 )=1 ab = c 2 a b a, b 2313 gcd(a, b, c) =1,a 2 + b 2 = c 2 a, b 2 = c 2 a 2 2314 b 2 =(c + a)(c a) (c + a), (c a) c + a = u 2, c a = v 2 (a, b, c) 2315,

23 79 231 x 1 > 1, x 1 2 t x 1,x 2,,x r (a) x 2 i + x2 i+1 (i =1, 2,,r 1) (b) p(x 1 ) p(x 2 ) p(x r )=2 x 2 + y 2 +1=z 2 232

80 24 10, 2, 360?, 17 36 50 18 26 10 4 26 60 0 24, 60 24 24 12 2 ( 0 1 ) 7? 0? 0?,?, 1 17 36 5841 5841 60 = 97 21 5841 97 21 97 97 24 = 4 1 4 1 17 36 4 1 21 18 57

24 81?, 2 8,1 12, 1 20 3 11 13 278,,? 30 12?? 30 31, 2 28 29? 1 364 364 = 7 52 1 52, 1,,, A, 2, 3,,Q,K 13? 13 4 1 (joker) 36524?? 3 1 n s(n) n : 1 2 3 4 5 6 7 s(n) : 1 3 6 10 15 21 28

82 x s(n) x<s(n +1) r = x s(n), x (n : r) x = s(n)+r, 10 (1:0)(1:1)(2:0)(2:1)(2:2)(3:0)(3:1)(3:2)(3:3)(4:0),25 (8 : 3) + 29 (1) s(6)=1+2+3+4+5+6=21, s(7) = s(6) + 7 = 28 s(6) 25 <s(7) 25 s(6) = 4, 25 = (6 : 4) (2) (8 : 3) + 29 = [(1 + 2 + +8)+3]+29=s(8) + 32 = s(8)+9+10+11+2=s(11) + 2 = (11 : 2)? n =0 s(n) =0, 0 =(0:0), 0 1 2 3 4 5 6 7 8 9 (0 : 0) (1 : 0) (1 : 1) (2 : 0) (2 : 1) (2 : 2) (3 : 0) (3 : 1) (3 : 2) (3 : 3)

24 83 4 (1) 37 (2)(10:1)+(5:4)? 5 1 n n! n : 1 2 3 4 5 6 7 n! : 1 2 6 24 120 720 5040 x n! x<(n +1)! k n! x<(k +1) n! k r = x k n!, x [k : n : r] 100 = 4 4! + 4 100 = [4 : 4 : 4] 300 [2:3:5]+67 (1) 5!=1 2 3 4 5 = 120, 6!=5! 6 = 720 5! 300 < 6! 2 5! 300 < 3 5! k =2 300 = 2 5! + 60 = [2 : 5 : 60] (2) [2:3:5]+67=(2 3! + 5) + 67 = 17 + 67 = 84, 84 84 = 3 4! + 12, 84 = [3 : 4 : 12]

84 6 (1) 1999 (2) [2 : 7 : 33] + [4 : 6 : 123] 5 3, 23 0, 12,24 12 7 8 8??? (, ) 8 91011121234, 4 (, )8+8=16 12 = 4 4

24 85 (, )8 8, 4 11 10 9 8 7 12 6 1 5 2 4 3 11 10 9 8 7 12 6 1 5 2 4 3 ( )? 8 10? ( ),, ( ) 12, 12 10 5 3 11 10 9 8 7 12 6 1 5 2 4 3 11 10 9 8 7 12 6 1 5 2 4 3 ( ), 0 11 1 12? 0 12 0

86 13 1,15 3, 7 5, a b 12?, a b 12? a b, c d ac bd 15? 5 15, 3 15 3 5 15 5, 15 12 12 15,? 15 15 17, 5 17 15 5 15 17 15 (3 15), 5 17 (5 17), 5 15 17 (3 5 15 17) 9 37 27? 37 1 (mod 12), 27 3(mod12) 37 27 1 3 = 3 (mod 12) 3 10 38 27?

24 87 38 27 2 27 2 4 =16 4=2 2,2 27 =2 4 2 23 2 2 2 23 = 2 25 2 25 2 23 2 3 =8 38 27 8 0 0 1 ( ) 1999 5 1999 0 0 0 5 1 1 0 5 0 5, 0 5 ( )? 1985 2 11 1999 10 4, 14 7 23 15, 1985, 1986, 1999 1985 15, 1999 12 25? 1999?, 1999 1998 ( ) 1 0, 1?0?, 1 1 0 1 31 174, 31 + 174 = 143 143 1 144

88 100, 1 100 1 100, 2 101 200, 20 1901 2000 2000 2001 1 1 1999 2000? 1 0? A, B, C A, B, C 1, 2, 3? A, B, C, AA, AB, AC, BA, BB, BC, CA, CB, CC, AAA, AAB,? 1, 2, 3, 4, 3?? 0 A B C AA AB AC BA BB BC CA CB CC AAA AAB 0 1 2 10 11 12 20 21 22 100 101 102 110 111? A, B, C 0, 1, 2 AA 00 3? 0 1, 1 A B C AA AB AC BA BB BC CA CB CC AAA AAB 1 2 10 11 12 20 21 22 100 101 102 110 111 112? A B 3 1, 2 C C?

24 89 3 10 AC A 1 C 10 20 A, B, C, 0, 1, 2 1, 2, 3 3 A B C AA AB AC BA BB BC CA CB CC AAA AAB 1 2 3 11 12 13 21 22 23 31 32 33 111 112 1 2 10 11 12 20 21 22 100 101 102 110 111 112? 4,5,10 0 0 11 1 12 11 2000 0?, 10 A 2000 = 1A00 = 19A0 = 199A 0 199A 2001 0 19A1? 1AAA 2111, 10 2110 2111,, 9999 10000 AAAA 11111,? 12 0 2111 1111 (!)? 0,

90 13 0 (1) 359A4 + A1234 = (2) 194 3A5 = 0 0? 0,0?

24 91 1 a 0,10 a 1,,10 n 241 a n, n +1? n n 1 242 n! a 0,a 1,a 2,? ( : 5 ) 1705 0?0 104 243 0? (1) i =1, 2, 3, 244 i(i 1) i(i +1) P i = x <x,x 2 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

92

3

94 31 n!? n! 1 9! 9!=1 2 3 4 5 6 7 8 9=2 1 3 1 2 2 5 1 (2 1 3 1 )7 1 2 3 3 2 =2 7 3 4 5 1 7 1 n! 1, 2, 3,,n, n n 1 n k k 2 100! 2, 100! = 2 a 3 b 5 c 7 d, a 100! 1 2, 3,, 100 100! 2 a a 2 6 2 1,4 2, 8 3

31 n! 95,4(=2 2 ) 2 a 1, 8( = 2 3 ) 4 a 2,2 4 2 3 a 3,,, 100 100 a = 1 2 1 2 2 +2 100 100 +3 2 3 2 4 + 100 100 = 1 2 1 +(2 1) 2 2 100 100 100 = 2 1 + 2 2 + 2 3 + 100 2 2 +(3 2) 100 2 3 100 2 3 +, 1 100 2 1,2 2,2 3,,? 1, 2,,100 2 2 ([100/2 1 ]) 4, 2 4 4 ([100/2 2 ]) 8 2 2 8 8 ([100/2 3 ]) 16 a = 100 2 1 + 100 2 2 + 100 2 3 + a =50+25+12+6+3+1=97

96 n!, p f(p, n) n n n f(p, n) = + p p 2 + p 3 +,, p k n 9! 1 9 2, 3, 5, 7 9!=2 a 3 b 5 c 7 d, 9 a = f(2, 9) = 2 9 b = f(3, 9) = + 3 9 c = f(5, 9) = =1 5 9 d = f(7, 9) = =1 7 9 + 2 2 9 3 2 9 + 2 3 =4+2+1=7 =3+1=4 3 20! 4 2000! 7?

31 n! 97 311 1004! a 10 n, n, a 1 0 10 n =2 n 5 n, 1004! 10 1004! = 2 a 3 b 5 c 7 d 2 a 5 c a c n 10 a c? 5 c, x y [x] [y] n 2 n k 5 k k n n n n n n 2 1 + 2 2 + 2 3 + 5 1 + 5 2 + 5 3 + n, n = 2000 5 1 + 2000 5 2 + 2000 = 400 + 80 + 16 + 3 = 499 5 3 + 3042! 8 0?

98 312 x x + + + 1! 2! x 2000! + = 1999 x 1999 x 1! x 1999 1999 < 7! = 5040, x x = a 6 6! + a 5 5! + a 4 4! + a 3 3! + a 2 2! + a 1, 0 a i i (i =1, 2,,7) 6! 1! + 6! 6! + + 2! 6! 5! a 6 + 1! + 5! 5! + + a 5 + + a 1 =1999 2! 5!, 1237a 6 + 206a 5 +41a 4 +10a 3 +3a 2 + a 1 = 1999 a 6 =0 1 (1) a 6 =0 : 206a 5 +41a 4 +10a 3 +3a 2 +a 1 (206+41+10+3+1) 5 < 1999 (2) a 6 = 1 : 206a 5 +41a 4 +10a 3 +3a 2 + a 1 = 762 41a 4 +10a 3 +3a 2 + a 1 41 4+10 3+3 2+1 = 201, 561 206a 5 762 a 5 =3 a 4 =3,a 3 =2,a 2 =0,a 1 =1, x =1 6!+3 5!+3 4!+2 3!+0 2! + 1 = 1165 x x x x + + + + + 1! 2! 3! 4! x 1001! + = 2002

31 n! 99 2000! 0? 311 100! 12, 0? 312 (a + b)!, a, b a! b! 313 314 1 2 2 2 2000 2,,, 2000 2000 2000?

100 x x x x + + + = 2000 1! 2! 3! 4! 315 2 100 n! n 316 (2m)! (2n)! m! n!(m + n)! 317 1 3 5 2001 9 0 318

32 101 32 2172 cm 2176 2171 cm?, 6519 3 2173 cm 1 n? n a 1,a 2,,a n A = 1 n (a 1 + a 2 + + a n ), cm, mm 10 10 10 t a i t A t, ta = 1 n (ta 1 + ta 2 + + ta n ) t

102, a 1,a 2,,a n a i, 2177, 2172 2176 2171, 4, 2172 2176 2171 4 6 2172, 4 1+1+4=6 a 1 w 1, a 2 w 2,, a n w n, W = 1 w 1 + w 2 + + w n (w 1 a 1 + w 2 a 2 + + w n a n ), 2?

32 103 3,,, a 1, a 2? l, r, w, wl = a 1 r, wr = a 2 l wlwr = a 1 ra 2 l lr w 2 = a 1 a 2, w = a 1 a 2 a 1 a 2 a 1 a 2 a n,? a i t, ta 1 ta 2 ta n = t n a 1 a 2 a n, ta 1,ta 2,,ta n a 1,a 2,,a n t n, n a 1,a 2,,a n G = n a 1 a 2 a n?

104 4 4 6 l? l 4 l 6? l 4 + l 6 l l 1 l 2 4 + l =48 6, 2ab 0 a b a + b a b, 0 n 1 + 1 + + 1 a 1 a 2 a n 5 (1) 3 5,? (2) A 2 B 8,? (3) 3 2,?

32 105 (1) (2) (3),,? a, b a + b 2 ab 2ab a + b, a = b? 1) ( ) ( ) a + b 2 ab = a + b 2 ab 2 = ( a b) 2 2 a + b ab 2 2) ( ) ( ) = ( a) 2 2 ab +( b) 2 2 0 ab 2ab a + b = = ab(a + b) 2ab ab(a + b 2 ab) = a + b a + b ab( a b) 2 0 a + b ab 2ab a + b

106 0 6 x + 1 (x>0) x x + 1 x 2 x 1 x =1, x + 1 x 2 x = 1 x x =1 2 a, b? n a 1,a 2,,a n 1,a n a 1 + a 2 + + a n n n n a 1 a 2 a n 1 + 1 + + 1 a 1 a 2 a n, a 1,a 2,,a n, a 1 = a 2 = = a n

32 107, 7 y = x + 1 (x>0) x y = x y = 1 x y = x + 1 x (y = x) (y = 1 x ) (y = x + 1 x ) x+ 1 2 x x + 1 2 x x =1 8 y = x + 4 x +1 -

108 (y = x) (y = 4 x+1 ) (y = x + 4 x+1 ) 3 -, x + 4 4 4 =(x +1)+ 1 2 (x +1) x +1 x +1 (x +1) 1=3 x +1= 4 x +1 x? x =1 x +1= 4, x =1 3 x +1,

32 109 321 a + b =8 ab (, a, b ) - a + b 2 ab, a + b 2 ab =16 2 a = b =4, ab a = b =4 16 (a + b) 2 = a 2 +2ab + b 2 =64 a 2 + b 2 =64 2ab (a b) 2 = a 2 2ab + b 2 0(1) 64 4ab 0, ab 16, (1) a = b, a = b =4, 16 a b + c b d a + d c ) 4 (, a, b, c, d

110 322 a, b, c 0 1 a + 1 b + 1 c 9 a + b + c - 3 a + b + c 3 3 1 a + 1 b + 1 b, 1 a + 1 b + 1 c 9 a + b + c a = b = c 3 - a, 1 b, 1 c 1 1 a + 1 b + 1 c 3 3 1/ 1 a +1/ 1 b +1/ 1 c = 3 a + b + c 3 10 m 2 10 10 m 2 20 20 m 2, 10 m 2 10 m 2, 20 m 2?,,

32 111 323 2( 2+2) a, b L L = a + b + a 2 + b 2 - a + b 2 ab a 2 + b 2 2ab (2 + 2) ab L =2( 2+2), ab 2 2 =4, S = 1 2ab 2 2, a = b = c PABC AP B = BPC = CPA =90 6 6( 2+1)

112 321 20 m? x 2+ 1 x 5 322 (x>5) x +3+ 1 2x 2 323 (x>1) a + nb (n +1) n+1 ab n (, a, b 0) 324 (a + b + c)(ab + bc + ca) 9abc (, a, b, c 0) 325

32 113 a, b, c, d abcd =1, 326 a 2 + b 2 + c 2 + d 2 + ab + ac + ad + bc + bd + cd 10 a, b, c (1 + a)(1 + b)(1 + c) =8 abc 1 327 a, b ab =1 k, 328 (k + a)(k + b) (k +1) 2 a, b, c abc =1 k 329, (k + a)(k + b)(k + c) (k +1) 3 a, b, c, d 3210 3 1 4 a + 1 b + 1 c + 1 1 d a + b + 1 a + c + 1 a + d + 1 b + c + 1 b + d + 1 c + d

114 1 321 a, b a + b ab x, y, z 322 x 2 + y 2 + z 2 =3 xy z + yz x + zx y

33 115 33 1 (1) 10, 50, 100, 500,12? (2) 3, 4, 5, 6? (1) 500 12 (2) 500 6, 100 5, 50 4, 10 3,,? 10, 50, 100, 500, 3, 4, 5, 6,, 3 10 + 4 50 + 5 100 + 6 500 4 10 + 6 50 + 3 100 + 5 500 2 a 2 + b 2 2ab 1 ( ) a 2 + b 2 2ab =(a b) 2 0

116 2 ( - ) (a 2 )+(b 2 ) 2 (a 2 )(b 2 )= ab ab 3 (!) a a + b b a b + b a a, b a, b (, ), (, ) ( ) a = b 3 a 2 + b 2 + c 2 ab + bc + ca (1) 2 (2) - (3)!, ( 2 ) a b, c d ac + bd ad + bc a = b c = d ac + bd ad bc = a(c d) b(c d) =(a b) (c d) 0 0 0

33 117 a 1 a 2 a n, b 1 b 2 b n {a k }, {b k } {a k}, {b k } ( ), a 1 b 1 + a 2 b 2 + + a n b n a 1b 1 + a 2b 2 + + a nb n a 1 b n + a 2 b n 1 + + a n b 1,,,,10, 50, 100, 500 6, 5, 4, 3 4 10 + 6 50 + 3 100 + 5 500 6 10 + 5 50 + 4 100 + 3 500, {a k } {a k},, a 1b 1 + a 2b 2 + + a nb n a 1 b 1 + a 2 b 2 + + a n b n {a n }, k } {b k }, {b k}, {a k } {a k }, {b k } {b k }, 2, a 1 b 1 + a 2 b 2 + + a n b n {b

118 b 1 <b 2, (a 1 b 2 + a 2 b 1)+ + a n b n (a 1 b 1 + a 2 b 2)+ + a n b n b 2,b 1,,b n b 1,b 2,,b n, i<j b i <b j {b k }, a 1 b 1 + a 2 b 2 + + a n b n, ( ), } {b k, i<j b i >b j a 1b 1 + a 2b 2 + + a nb n a 1 b n + a 2 b n 1 + + a n b 1,, 4, 2, 3, 1,, 4, 2, 3, 1 2, 4, 3, 1 2, 3, 4, 1 2, 3, 1, 4 4 4,

33 119 2, 3, 1, 4 2, 3, 1, 4 2, 1, 3, 4 3 2, 1, 3, 4 1, 2, 3, 4 (bubble sort) (quick sort), (sample sort),,oa 1980 a 1 a 2 a n = a 1 b 1 + a 2 b 2 + + a n b n b 1 b 2 b n

120, ( ; ) a 1 a 2 a n, b 1 b 2 b n {a k }, {b k } {a k}, {b k }, a 1 a 2 a n b 1 b 2 b n a 1 a 2 a n b 1 b 2 b n a 1 a 2 a n b n b n 1 b 1 4 a 3 + b 3 + c 3 a 2 b + b 2 c + c 2 a, a, b, c 0 a 3 + b 3 + c 3 = a2 b 2 c 2 a2 b 2 c 2 a b c b c a = a 2 b + b 2 c + c 2 a 5 a 2 + b 2 a + b 2 2 2

33 121 a 2 + b 2 2 a + b 2 ( )-( ) 1 ( ) 4 (a b) 2 0 a = b 2 ( - ) 4 - a 1 = b 1 (1 2 +1 2 )(a 2 + b 2 ) (1 a +1 b) 2 (,!) a, b,, {a, b} ( ), a 2 + b 2 ( ) aa + bb ( ), ( a+b a+b 2 )( 2 ( ) ) a + b 2 a b + b a (= ab) 2 2 -,

122 3 a 2 + b 2 = a 2 + b 2, a 2 + b 2 a b + b a, 2(a 2 + b 2 ) (a + b)(a + b) 4 (Chebyshev) {a k } {b k ( ), a 1 b 1 + + a n b n n a 1 + + a n n b1 + + b n n a 1b n + + a n b 1 n a 1 b 1 + + a n b n = a 1 b 1 + a 2 b 2 + + a n b n, a 1 b 1 + + a n b n a 1 b 2 + a 2 b 3 + + a n b 1, a 1 b 1 + + a n b n a 1 b 3 + a 2 b 4 + + a n b 2, a 1 b 1 + + a n b n a 1 b n + a 2 b 1 + + a n b n 1 n(a 1 b 1 + + a n b n ) (a 1 + + a n )(b 1 + + b n ) n 2

33 123 6 a, b > 0, a 2 + b 2 a + b a + b 2,, a a + b b 2 a + b 2 a + b 2 7 a, b, c > 0, a 3 + b 3 + c 3 a 2 + b 2 + c 2 a + b + c 3

124 331 (,Nesbitt) a, b, c a b + c + b a + c + c a + b 3 2 a b + c +1= a + b + c b + c, 3 1 (a + b + c) a + b + 1 b + c + 1 9 c + a 2 x = a + b, y = b + c, z = c + a, x + y + z =2(a + b + c) 1 2f(a, b, c) =(x + y + z) x + 1 y + 1 9 z 1 1 2f(a, b, c) =(x + y + z) x + 1 y + 1 =3+ x z y + y + y x z + z + z y x + x 9 z 2 2 2 -, y = y x, y z = z y, z x = x z x = y = z, a = b = c x 2 ( u + v + w 3 - ) 3 1 u + 1 v + 1 w (u + v + w) 1 u + 1 v + 1 w 9 u, v, w x, y, z, x = y = z, a = b = c

33 125 3 ( - ) u + v + w 3 3 uvw 1 2f(a, b, c) =(x + y + z) x + 1 y + 1 3 z 3 1 xyz 3 3 xyz =9 4 (!), a, b, c a b c b + c c + a a + b, 1 b + c 1 c + a 1 a + b, a b c 1 c + a 1 b + c 1 b + c 1 a + b a b c 1 c + a 1 a + b b c a 1 c + a 1 b + c 1 b + c 1 a + b c a b 1 c + a 1 a + b a b c b + c c+ a a+ b 2 1 1 1 1 1 1 =1+1+1=3 b + c c + a a + b b + c c + a a + b,, (1990 ) a i > 0(i =1,,n) s = a 1 + + a n, a 1 s a 1 + a 2 s a 2 + + a n s a n n n 1

126 332 m, n, m m n n m n n m (m m) (n n) (m m) (n n) m n n m, m n m + n,,, m n n n m n (m n n ) m m n n m n m n, n m ( ) m = n a, b, a a b b a b b a

33 127 331 a a b b c c (abc) (a+b+c)/3, a, b, c 0 <a b c d, 332 a b b c c d d a b a c b d c a d a, b, c > 0, 333 a + b + c abc 1 a 2 + 1 b 2 + 1 c 2 x i > 0(i =1,,n), s = x 1 + + x n 334 s s x 1 + s s x 2 + + s s x n n2 n 1 x, y, z > 0, 335 x 2 y 2 + y2 z 2 + z2 x 2 y x + z y + x z

128 x, y, z > 0, 336 x 2 y 2 + y2 z 2 + z2 x 2 x y + y z + z x a, b, 337 (a + b)(a 4 + b 4 ) (a 2 + b 2 )(a 3 + b 3 ) a, b, c, 338 a n b + c + bn c + a + cn a + b an 1 + b n 1 + c n 1 2 a, b, c > 0, 339 abc(a + b + c) a 3 b + b 3 c + c 3 a (2002 ) 3310 a 4 + b 4 + c 4 a 2 bc + b 2 ca + c 2 ab

33 129 0 <a b c d, 331 a b b c c d d a b a c b d c a d (1975 ) {x i }, {y i } (i =1,,n) 332, {z i } {y i } n (x i y i ) 2 i=1 n (x i z i ) 2 i=1, n i=1 a i = a 1 + a 2 + + a n (1978 ) {a i } (k =1, 2,,n,) 333 (, ) n, n k=1 a k k 2 n k=1 1 k

130, 334? a 1 a n b 1 b n a 1 a n b 1 b n x 1 x n x 1 x n, {a k }, {b k },,{x k }, {a k }, {b k },, {x k }

4

132 41 ABC BC M, AB 2 + AC 2 =2(AM 2 + BM 2 ) 1 A BC H 1 P AB, BC, CD, DA PE, PF, PG, PH PE PG = PF PH

41 133 PEB PHD ABP = PDA PEB PHD, PE PH = PB PD PBF PDG PBF = PDG PBF PDG, PG PF = PD PB PE PH PG PF =1 PE PG = PF PH 2 O OA, OB O P OA, OB, AB E, G, F PF 2 = PE PG

134 E = G = AF P = PFB = R EAFP, PFBG 180 BP BG PBG BP PAF, PFG = PBG = PAF = PEF PFE = PAE = PBF = PGF

41 135 PEF PFG, PE : PF = PF : PG PF 2 = PE PG (Van Aubel, 1881 ) ABC O AO, BO, CO D, E, F AO DO = AF BF + AE CE A, B CF AF : BF AOC BOC OC AOC BOC = AF BF AOB BOC = AE CE

136 AOC + AOB BOC = AF BF + AE CE AO DO = AOB BOD = AOC COD = AOB + AOC BOD + COD = AOB + AOC BOC AO DO = AF BF + AE CE 2 (Gergonne, 1818 ) ABC O, AO, BO, CO D, E, F OD AD + OE BE + OF CF =1 (Monge, 1764 1818 ) A, B, C

41 137 A B, B C, C A PQ, RS, TU, RS, TU O OP A, B Q, Q C A RO OS = TO OU, TO OU = PO OQ, B RO OS = PO OQ PO OQ = PO OQ OQ = OQ, Q Q PQ O 3 (Carnot, 1803 ) ABC AB, BC, CA D D, E E, F F AD BE CF BD CE AF AD BE CF BD CE AF =1

138

41 139 411 (1987 ) C =90 ABC AB M, C AB H, CH, CM C, CHM : ABC 90 1 3 30, 4, CHM 4 1 :4 1 ABC CB CA a, b, C = 120 C a b

140 412 AD//BC ABCD O BOC = p 2, AOD = q 2, ABCD (p + q) 2 BOC AOD p 2 : q 2 p : q, BO : DO = CO : AO = p : q BOC COD C, BOC : COD = BO : DO = p : q, COD = pq AOB = pq, ABCD p 2 + q 2 +2pq =(p + q) 2 ABC AB C 1, C 1 C A C 1 C BC A 1,, B C 1 C AC B 1 1 AA 1 + 1 BB 1 = 1 CC 1

41 141 413 ABC BC D, E BAD = CAE, AB 2 : AC 2 = BD BE : CD CE A, D, E AB, AC P, Q PAD = QAE PD = QE, PDB = PAE = QAD = QEC PQ BC, AB : AC = AP : AQ = BP : CQ AB 2 : AC 2 = AB BP : AC CQ, A, P, D, E, AB BP = BD BE, AC CQ = CD CE, AB 2 : AC 2 = BD BE : CD CE O AB TD(D ) C AB, E C TD, AC CB + CD 2 = CE AB

142 ABC BC, CA, AB D, E, F, 411 AD 2 + BE 2 + CF 2 = 3 4 (BC2 + CA 2 + AB 2 ) AB AC O B, C BD 412 CE BE BO = AB CE 6 O CD O 3 413 CD( ) A AB, CD M B AB AM ABCD A BD, CD, 414 BC P, Q, R, PD 2 : PB 2 = PQ : PR

41 143 415 O D, A, C, B, AB, CD A C P AB, CD E, F, PE : PF = AB : CD A ABCD O 416 BC E, OE AB F, BE (AB +2BF) =BC BF ABCD ABC = DCB, DA 417 CB P, PA PD = PB PC + AB CD ABC A BC D, 418 E, AB AC AD AE =1

144 (1991 - ) ABC 419 G, BC M, G BC AB, AC X, Y GC, YB P, GB, XC Q, MPQ ABC (1996 - ) ABCD 4110 AB, BC, CD, DA E, F, G, H, EF HG BD EF HG AEF CGH AB AC, 4111 AC AC AB D AB = 10, AD : DB =2:3, AC? ABCD AC C 4112 AB AD E F, AB AE + AD AF =(AC) 2

41 145 (1979 ) A 411 P P B, C 1 BP + 1 BC PC ABC P BC, CA 412 D, E AD, BE L, M, DE, LM ABCD M AD//BC, 413 AB AD R DCM

146 42, BC 375, (1) (2) 1 AB A AB B AB C

42 147 AC AB A, BC ABC 2 A 3 P, P 4 P, P,,,,, O AB

148, A O O A D E, D B E B F DE OF = AB, O F AB 5 AB A, B AB X, Y, AB M, 6 AB 7 A

42 149 A, A C D C D AD DC P DC A AP, SSS ABC ACP, AP 8 A A B A AB A C OX O AB D D BC, O E AB = AC = OD = OE, BC = DE ABC ODE A = EOD 9 P 10 O A A O

150 OA O, OA O O O B, C, OBA O AB O AB AC 1 a, b, a a + b, a b, a b, b, a, n,,, 11 AB 5 A AA 1 = A 1 A 2 = A 2 A 3 = A 3 A 4 = A 4 A 5, A 5 B

42 151 A 5 B AB 5 12 m n m m : n, 13 A, B, C O O (!), O A, B, C ( ) AB, BC A, B, C O,

152 ( ) 1, 2 3 4 5,, 6, ( ) 7 8 9

42 153 14 1, 2, 3 1, 2, 3 (1 + 2), (2 + 3), (3 + 1) A, B, C 2, 1, 3 15 16 17 18

154 19 15? 4, 3, 4, 5, 6, 15,3 2 n,4 2 n,5 2 n,15 2 n n 1796, 19 f(n) =2 2n +1 n =0, 1, 2, 3, 4 2, 5, 17, 257, 65537 f(5) 641, 5 n 19 f(n) 17, 257 1832 (F J Richelot), 65537 10, 7, 9, 11, 13 19

42 155 421 1, a, b, a b AB = a + b, AD = a, BD = b, CD =1 ABC CD E AD BD = CD DE DE = a b (1) 1, a, b, b (2) 1, a, a a

156 422 A, B C ACB C A, B T, ACB C T A, B AB P PT 2 = PA PB PA PB P T

42 157 423, C 1, C 2 C 1, C 2, C 2?( ) C 1, m m C 2, n n n m P, C 2 n, P C 2 C2, C 1, C2 : C 2, C 1, C 2

158 424 ABC, BC, ( ) AB D BC E DEFG, G AC, BG AC G, BC F B D E F G

42 159 421 O n, 2n 422 60 3 n 3 423 360 n 424 P, Q, R P, Q, R 425 A, B, C C 426 A B

160 7 427, ABC 8 428,,,, 1 A ABC, AB, AC D, E 429 AD = DE, AE = DB AB c, r, AB 4210 r c ABC A, B, C A ( 4211 ) B C

42 161 n = ab 421 n a b a, b a, b ab 422 423 424 A, B, C PQR A, B 425, C PQR

162

5

164 51 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1??, 1 (1)? (2)? (3)? (4),? (1) 1 (2) 1, 2, 3, 4, 5, i i (3)1,2,3,4 1, (4)

51 165 2 (5)? 1 = 1 = 2 0 1+1 = 2 =2 1 1+2+1 = 4 =2 2 1+3+3+1 = 8 =2 3 1+4+6+4+1 = 16 =2 4? i 2, 2 i 1 (1) (3) 1 5 5 1 (4), (5) 1+5+ + +5+1=2 5 =32 =10, 6 1 5 10 10 5 1 7? 1 6 6 1 x, y x +2y + 14 = 64, x +2y =50

166 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 6 1! 3+3 4+6 10?? 1 1, 1 0, 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 2 1 0 0 0 1 3 3 1 0 0 1 1, 0 1,

51 167 3 7,8,9 6 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1,?, 4 7,8,9 2? 7 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64 = 2 6 8 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = 128 = 2 7 9 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256 = 2 8 2? 2? 1 3 3 1 /\ /\ /\ /\ 1 4 6 4 1

168, 1+4+6+4+1 = (1)+(1+3)+(3+3)+(3+1)+(1) = (1 + 1) + (3 + 3) + (3 + 3) + (1 + 1) = 2(1+3+3+1)=2 2 3 =2 4, 0 0, n r C n,r (A) C 0,0 = C n,0 = C n,n =1 (B) C n,r + C n,r+1 = C n+1,r+1 (C) C n,0 + C n,1 + + C n,n =2 n (A) 1,(B),(C) 2 5 C n,2 n C n,2 n 2 1, 3, 6, 10, 15, 21, 28, 1, 1,2, 3, 4, 5,

51 169 (A) C n,0 1, (B) C n,1 = C n 1,1 + C n 1,0 = C n 1,1 +1, C n,1 = C n 1,1 +1=C n 2,1 +1+1= = C 1,1 +1+ +1 n 1 =1+(n 1) = n C n,0, C n,1 C n,2 (B) C n,2 = C n 1,2 + C n 1,1 = C n 1,2 +(n 1), C n,2 = C n 1,2 +(n 1) = C n 2,2 +(n 2) + (n 1) = = C 2,2 +2+ +(n 1) = 1 + 2 + +(n 1) C n,2 = n(n 1) 2 6 nc r n r nc r = n (n 1) (n r +1) r 2 1 = n! r!(n r)! n =3, 4, 5 nc r

170 n =3 3C 0 =1, 3C 1 = 3 1 =3, 3C 2 = 3 C 1 =3, 3C 3 =1 n =4 4C 0 =1, 4C 1 =4, 4C 2 = 4 3 2 1 =6, 4C 3 = 4 C 1 =4, 4C 4 =1 n =5 5C 0 =1, 5C 1 =5, 5C 2 = 5 4 2 1 =10, 5C 3 =10, 5C 4 =5, 5C 5 =1 3C 03 C 13 C 23 C 3 4C 04 C 14 C 24 C 34 C 4 5C 05 C 15 C 25 C 35 C 45 C 5 = 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1?,? nc r nc 0 = n C n =1, n+1c r+1 = n C r + n C r+1,

51 171 n C 0 = n C n =1 nc r = n! r!(n r)! r =0 r = n nc r + n C r+1 = = = n! r!(n r)! + n! (r +1)!(n r 1)! n! [(r +1)+(n r)] (r +1)!(n r)! (n +1)! (r +1)!(n r)! = n+1c r+1,, n C n,r nc r n +1, nc r n r n +1 A = {1, 2,,n+1} r +1, n +1 n +1 n r nc r, n +1 n r +1 nc r+1 r+1 r r+1 n+1 n 1 1 n 0 1, n+1c r+1 = n C r + n C r+1

172,,

51 173 511 1 1 1 1 1 2 3 2 1 1 3 6 7 6 3 1 (1) (2) n,, 0 (3) (1) 1,, 1 4 10 16 19 16 10 4 1 (2),, 3 0 ( ) 1, n 3 n (3),,, 1, x \ / y

174, x y = x +2,, (3) (2), 3 n, 111 1 1 1 1 2 2 1 1 3 4 3 1 0, n,, 0, n 2 n

51 175 512 1 1 3 3 1 3 1 2 1 3 3 2 1 1 2 3 1 1 1 1 1 1 1 3 2 1 1 2 3 3 1 2 1 3 1 3 3 1 1 0, 1 1, 1 2 2, (1) n, (2) 1 6 2 1 n, n (1) 6 n n, 2 n 6,,, 1, 6 2 n 6

176, n 1, n =0 1 (2) n 2n, 2n C n, 6 2n C n = 6 (2n)! (n!) 2 1 1 3 6 7 6 3 1 3 1 2 3 2 1 3 6 2 1 1 1 2 6 7 3 1 1 1 3 7 6 2 1 1 1 2 6 3 1 2 3 2 1 3 1 3 6 7 6 3 1 n,, 1 0

51 177 511 (n>3) 512 n+1c 1 + n+1 C 2 + n+1 C 3 < 2( n C 1 + n C 2 + n C 3 ) 513 1 1 1 1 1 2 3 2 1 1 3 6 7 6 3 1 0, 0, n r D n,r (1) D n,1 n (2) D n,2 n (1988 ) 514

178 111 515 1 1 1 1 2 2 1 1 3 4 3 1 (2001 KAIST ) 516, 0 1 1 2 2 2 3 4 4 3 4 7 8 7 4 5 11 15 15 11 5 6 16 26 30 26 16 6 7 22 42 56 56 42 22 7 a, b D(a, b), D(a, b), 0, n

51 179 511 3 1 1 1 1 1 2 1 2 2 1 1 3 3 1 3 6 3 3 3 1,, nc r = n! r!(n r)!,, n 512? k k 513 0, 1, 3, 7, 15,, 2 n 1,

180 52,, ( ),? ( )! 1 2 3 4 A B C D ( )! ( ),!,, 1 2 3 4 ( )? ( ) 3? ( ) 4, 1 1 2 3 4

52 181 ( ) 2??, ( ),? ( ), ( ),!, ( ), ( ),?? ( )?,? ( )!,? ( ), 1 2 3 4 1 2 3 4 A B C D A B C D ( ),!,? ( ),

182 1 2 3 4 A B C D, X-? ( ), X- 2 3 1 2 3 4 1 3 2 4 A B C D A B C D ( ), X- 1 2 3 4 1 2 3 4 ( ),! 4 ( ),? X-?

52 183 1 2 3 4 ( ), ( ),? 1 1?,? ( ),??? L M N

184 L X- X-, ( ) M??? ( ),? N, ( )? ( ) X- X- 1 2 3 4 3 2 4 1 ( ), X-??? ( )?

52 185?? ( ),, 3 2 4 1 ( ) ( ), ( ) X-,? ( )? ( )

186,,,!?,?,? 1? 1 2 3 4 1 2 3 4 A B C D A B C D??,?,

52 187 1, 2, 3, 4 1 2 3 4 1 D,4 D A 2 B 3 3 A A 4 3 4 A B C D 1 2 3 4 1 D, A B C D 1, 2, 3, 4, 3, 4, 1, 2,,, A, B, C, D,

188 2, A, B, C, D 3,???,,, 4? 1 2 3 4 A B C D

52 189 521 :,, 522? 523,, 524 15-1 2 3 4 5 6 7 8 9 10 11 12 13 15 14-1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

190