x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy x y) = 3 5 x, y x + y = 3, 5, 15 ( i ) x + y = 3 (x y) (xy 3) = 5 (x y, xy 3) = (1, 5), (5, 1), ( 1, 5), ( 5, 1) (x y, xy) = (1, 8), (5, 4), ( 1, 2), ( 5, 2) ( ii ) x + y = 5 (x y) (xy 5) = 3 (x y, xy 5) = (1, 3), (3, 1), ( 1, 3), ( 3, 1) (x y, xy) = (1, 8), (3, 6), ( 1, 2), ( 3, 4) x = 1, y = 4 (iii) x + y = 5 (x y) (xy 15) = 1 (x y, xy 15) = (1, 1), ( 1, 1) (x y, xy) = (1, 16), ( 1, 15) -1-
2 a2 + b 2 + 2 2 = a + b + 2 a, b 1977 a2 + b 2 + 2 2 = a + b + 2 a 2 + b 2 + 2 2 = (a + b + 2) 2, a + b + 2 2 2ab + 4a + 4b = 0, a + b 0 (a + 2) (b + 2) = 4, a + b + 4 4 {a + 2, b + 2} = {1, 4}, {2, 2} {a, b} = { 1, 2}, {0, 0} 3 m 6x + my = 15 x y y 1977 6x + my = 15 my = 12 6x + 3 my = 6M + 3 m m 6 ( i ) m = 6a + 1 y = 6b + 3 y = 3 x = 5 m 2 ( ii ) m = 6a + 3 y = 6b + 1 y = 1 x = 15 m 6 (iii) m = 6a + 5 y = 6b + 3 y = 3 x = 5 m 2 4 n n n n 2 1977 x 1 + x 2 + + x n = x 1 x 2 x n 1 x 1 < x 2 < < x n -2-
nx n x 1 + x 2 + + x n = x 1 x 2 x n (n 1)! x n n (n 1)! 2 1 (: n = 2), 3 2 (: n = 3), 4 6 (: n = 4) n 4 n < (n 1)! n! > n 2 > n + 1 n 4 n (n 1)! n = 2, 3 n = 2 x 1 + x 2 = x 1 x 2 (x 1 1) (x 2 1) = 1 x 1 1 = x 2 1 = 1 n = 3 x 1 + x 2 + x 3 = x 1 x 2 x 3 1 = 1 + 1 + 1 3 x 1 x 2 x 2 x 3 x 3 x 1 x 2 1 x 2 1 3 x 1 = 1 1 + x 2 + x 3 = x 2 x 3 (x 2 1) (x 3 1) = 2 x 2 1 = 2, x 3 1 = 2 x 2 = 2, x 3 = 3 5 p, q x 2 + px + q = 0 1977 x 2 + px + q = 0 x = 2m 4m 2 + 2pm + q = 0 x = 2m + 1 4m 2 + 4m + 1 + 2pm + p + q = 0-3-
6 p 100 a,10 b,1 c ax 2 + bx + c = 0 1977 p = 10 2 a + 10b + c 1 ax 2 + bx + c = 0 x = n an 2 + bn + c = 0 2 1-2 p = a (10 n) (10 + n) + b (10 n) = (10 n) {a (10 + n) + b} 2 n < 0 10 n 11 10 n = p, a (10 + n) + b = 1 n a (20 p) + b = 1 0 b 9 0 1 a (20 p) 9 1 a (p 20) 8 3 p 80a a (p 20) 979a 4 3 4 7 x 4x 3 (a 2)x (a + 4) = 0(a ) 1977 4x 3 (a 2) x (a + 4) = 0 x = q p (p, q N, p, q ) -4-
( ) 3 q 4 (a 2) q (a + 4) = 0 p p 4q 3 (a 2) p 2 q (a + 4) p 3 = 0 4q 3 = p 2 {(a 2) q + (a + 4) p} 4q 3 p 2 p, q p 2 4 p = 2 q 3 = (a 2) q + 2 (a + 4) = a (q + 2) 2q + 8 a = q3 + 2q 8 q + 2 q + 2 = 4, 5, 10, 20 q = 2, 3, 8, 18 = p 2 2p + 6 20 q + 2 q = 3 x = 3 2 8 x y ( ) 2, 1 (1) 3 ( ) 2, 1 (2) 1 A, B, C 3 (3) n n 1977-5-
(1) y (a, b), (c, d) (x a) 2 + (y b) 2 = (x c) 2 + (y d) 2 2 (c a) x + 2 (d b) y = c 2 + d 2 a 2 b 2 ( ) 2, 1 3 2 1 C 2 (c a) 2 + 2 (d b) 1 3 = c2 + d 2 a 2 b 2 P c = a 2 (d b) 1 3 = d2 b 2 O A 1 2 B x d \= b d + b = 2 3 c \= a c 2 + d 2 a 2 b 2 2 2 = 3 (d b) 2 (c a) (a, b), (c, d) (2) A(1, 0),B(2, 0),C(1, 1) ( ) 2, 1 3 (3) P A 1,A 2,A 3, PA k = r k (k = 1, 2, 3, ) r n,r n+1 r n < r < r n+1 r P r n 9 n (i),(ii) ( i ) n ( ii ) 1 < p n p n p 1977 n 1 < p n p n p -6-
n n = p 2 Q 1 (p, Q 1 ) 1 n = pqq 2 (p, q, Q 2 ) 2 1 (ii) 2 p < q p 2 < pqq 2 (ii) 10 m, n 2m + 3n m + 2n 1977 2m + 3n, m + 2n p( 2) { 2m + 3n = pk m + 2n = pj { m = p (2k 3j) n = p (2j k) m, n p 11 1x100y 38 x, y 1977 1x100y (10) = 10 5 + x 10 4 + 10 3 + y 22 + 6x + 12 + y (mod38) 6x + y 4 (mod38) 3 6x + y 4 59 6x + y 4 = 38 6x + y = 42 y 6 y = 6, x = 6-7-
12 n n 2 + n 10 T (n) p, q, t (1) T (3), T (4), T (5), T (6) (2) p + q = 9 T (p) = T (q) (3) n T (n + t) = T (n) t 1977 (1) n 3 4 5 6 n(n + 1) 12 20 30 42 T (n) 3 0 0 2 (2) p 2 + p ( q 2 + q ) = (p q) (p + q + 1) = 10 (p q) T (p) = T (q) (3) T (n + t) = T (n) (n + t) 2 + n + t ( n 2 + n ) = 10M 2nt + t 2 + t = 10M n t 2 + 3t = 10M 1 (n = 1) t 2 + 5t = 10M 2 (n = 2) 2t = 10M 3 t = 5M 3 t 5-8-
13 (1),(2) (1) x 0, x 1, x 2, x 3 3 (2) n n (n + 1) 33 300 0 1977 (1) x 0, x 1, x 2, x 3 0,1,2 x 0, x 1, x 2, x 3 (2) 3, 33, 333,, 333 3 }{{} n+1 n + 1 n 33 300 0 n n + 1 14 (1) p p 3 + (p + 1) 3 + (p + 2) 3 9 (2) p > 3 p p + 2 p + 1 6 1977 (1) P = p 3 + (p + 1) 3 + (p + 2) 3 = 3p 3 + 9p 2 + 15p + 9 = 3p 3 3p + 9 ( p 2 2p + 1 ) = 3 (p 1) p (p + 1) + 9 ( p 2 2p + 1 ) (p 1) p (p + 1) 3 P 9 (2) p (p + 1) (p + 2) 3 6 p p > 3 p + 1 6-9-
15 6 = 0 2 + 1 2 + 1 2 + 2 2 7 = 1 2 + 1 2 + 1 2 + 2 2 30 = 1 2 + 2 2 + 3 2 + 4 2 = 0 2 + 1 2 + 2 2 + 5 2 (1) 21,43 (2) l 2 + m 2 + n 2 = 23(0 l m n) l, m, n 1977 (1) 21 = 4 2 + 2 2 + 1 2 + 0 2 43 = 5 2 + 4 2 + 1 2 + 1 2 (2) l 2 + m 2 + n 2 = 23 (0 l < m n) l, m, n 23 = l 2 + m 2 + n 2 3l 2 l = 0, 1, 2 l = 0 m 2 + n 2 = 23 m = 1 2 n 2 = 22 ( ) m = 2 2 n 2 = 19 ( ) m = 3 2 n 2 = 14 ( ) m = 4 2 n 2 = 7 ( ) l = 1 m 2 + n 2 = 22 m = 1 2 n 2 = 21 ( ) m = 2 2 n 2 = 18 ( ) m = 3 2 n 2 = 13 ( ) m = 4 2 n 2 = 6 ( ) -10-
l = 2 m 2 + n 2 = 19 m = 1 2 n 2 = 18 ( ) m = 2 2 n 2 = 15 ( ) m = 3 2 n 2 = 10 ( ) l 2 + m 2 + n 2 = 23(0 l m n) l, m, n n, a, b, c, d 0 a 2 + b 2 + c 2 + d 2 = n 2 6 a + b + c + d = n a b c d 16 (n, a, b, c, d) 1980 (a + b + c + d) 2 = a 2 + b 2 + c 2 + d 2 + 2 (ab + ac + ad + bc + bd + cd) n 2 = n 2 6 + 2 (ab + ac + ad + bc + bd + cd) ab + ac + ad + bc + bd + cd = 3 3 6d 2 d = 0 ab + ac + bc = 3 3 c 2 c = 0, 1 c = 0 ab = 3 (a, b) = (3, 1) c = 1 ab = 2 (a, b) = (2, 1) (n, a, b, c, d) = (4, 2, 1, 1, 0), (4, 3, 1, 0, 0) -11-
17 n p 3 x 3 + nx 2 (5 n)x + p = 0 1980 x 3 + nx 2 (5 n) x + p = 0 m 1 m 3 + nm 2 (5 n) m + p = 0 m { m 2 nm + (5 n) } = p m = p, m 2 nm + (5 n) = 1 1 m = 1, m 2 nm + (5 n) = p 2 p 2 np + (5 n) = 1 p 2 + np + n = 4 p 2 + np + n 2 2 + 2 + 1 2 1 n + (5 n) = p p = 4 2n p = 2, n = 1 x 3 + nx 2 (5 n) x + p = 0 x 3 + x 2 4x + 2 = 0 (x 1) ( x 2 + 2x 2 ) = 0 v = 1, x = 1 ± 3-12-
18 0,1 a, b(a \= b) f(x) = x(x 1)(x a)(x b) + 1 g(x), h(x) f(x) = g(x)h(x) (1) g(0) = h(0) (2) g(x), h(x) g(x) = h(x) (3) (2) a, b 1980 f (x) = x (x 1) (x a) (x b) + 1 = g (x) h (x) g(x), h(x) n f(n), g(n) (1) f (0) = 1 = g (0) h (0) g (0) = h (0) = 1, g (0) = h (0) = 1 (2) g(x), h(x) g(x), h(x) f (1) = 1 = g (1) h (1) f (a) = 1 = g (a) h (a) f (b) = 1 = g (b) h (b) (1) g (1) = h (1), g (a) = h (a), g (b) = h (b) g (x) = h (x) -13-
(3) y g(x), h(x) 3 x (x 1) (x a) (x b) + 1 = ( x 2 cx + 1 ) 2 2 x (x 1) (x a) (x b) + 1 = ( x 2 cx 1 ) 2 g(x) = x 2 cx + 1 y = g(x) y = 1,y = 1 1 1 O 1 1 2 3 4 x x 0, b; 1, a a = 2, b = 3 19 a, b f(x) = x 2 + ax + b f(α) = 0 α (1) α (2) l n n f(l), f(l + 1),, f(l + n 1) n 1980 (1) α = p (p, q Z, ) q ( ) 2 ( ) p p + a + b = 0 q q p 2 + apq + bq 2 = 0 p 2 = q (ap + bq) q p 2 p, q q = 1 α (2) l α + k(k = 0, 1, 2,, n 1) n r(r = 0, 1, 2,, n 1) l + k = α + nq + r f (l + k) = f (α + nq + r) = (α + nq + r) 2 + a (α + nq + r) + b (α + r) 2 + a (α + r) + b (mod n) = f (α) + 2αr + r 2 + ar = r (2a + α + r) r = 0 k f (l + k) n -14-
20 n, p n p n p+4 1981 n p+4 n p = n p ( n 4 1 ) = n p { (n 2) (n 1) (n + 1) (n + 1) + 5n 2 5 } = n p 1 (n 2) (n 1) n (n + 1) (n + 1) +5 (n 1) n (n + 1) }{{}}{{} (A) (A) 5! = 120 (B) 3 3! = 6 n p+4 n p 10 n p n p+4 (B) 21 f(x) = ax 3 + bx 2 + cx + d n f(n) f(x) 1981 f (n) = an 3 + bn 2 + cn + d n f (n + 1) f (n) = a ( 3n 2 + 3n + 1 ) + b (2n + 1) + c 1 {f (n + 2) f (n + 1)} {f (n + 1) f (n)} = f (n + 2) 2f (n + 1) + f (n) = a (6n + 6) + 2b 2 {f (n + 3) 2f (n + 2) + f (n + 1)} {f (n + 2) 2f (n + 1) + f (n)} = f (n + 3) 3f (n + 2) + 3f (n + 1) f (n) = 6a 3 3 6a 2 2b 1 a ( 3n 2 + 3n + 1 ) + b (2n + 1) + c = 3a n (n + 1) +2b n + a + b + c }{{} -15-
n a + b + c 6a + 6b + 6c 6c f(0) = d 6d 22 m 2 = 2 n + 1 m, n 1982 m 2 = 2 n + 1 m m = 2M + 1 (2M + 1) 2 = 2 n + 1 4M 2 + 4M = 2 n M (M + 1) = 2 n 2 M, M + 1 1 M = 1 M + 1 = 2 n = 3 M + 1 = 1 M = 0 m = 3, n = 3 23 n + 7 5 n + 5 n 1982 n + 7 = 5k n + 5 = 7j 5k 7j = 7 5 5(k + 1) = 7(j + 1) 7 k + 1 = 7m, j + 1 = 5m n = 5 (7m 1) 7 n 23 24 a, b a + 2 b b + 1 a a, b 1982-16-
a + 2 = bm 1 b + 1 = an 2 1 2 a 2b = bm 2an a (2n + 1) = b (m + 2) 2 a, b a = m + 2, b = 2n + 1 (m + 2) + 2 = (2n + 1) m 2mn = 2 + 2 mn = 2 (m, n) = (2, 1), (1, 2) (a, b) = (4, 3), (3, 5) 25 x, y a = 5x + 4y, b = 6x + 5y (1),(2) (1) a, b x, y (2) 4 5 < r < 5 6 r x, y r = a b 1982 (1) { a = 5x + 4y { x = 5a 4b b = 6x + 5y y = 6a + 5b x, y G,a, b K x = Gx, y = Gy (x, y ) { a = G (5x + 4y ) b = G (6x + 5y ) G a, b G K 1 a, b K a = Ka, b = Kb -17-
{ x = K (5a 4b ) y = K ( 6a + 5b ) K x, y K G 2 1, 2 G = K (2) 4 5 < r < 5 6 r r = a b 4 5 < a b < 5 5a 4b > 0, 6a + 5b > 0 6 { x = 5a 4b y = 6a + 5b x, y x, y a, b { a = 5x + 4y b = 6x + 5y 26 (1) (2) 1982 (1) a, b, c, d, e(a b c d e) abcde = a + b + c + d + e 0 < a b c d e abcd > 5 abcde > 5e a + b + c + d + e > 5e (e a) + (e b) + (e c) + (e d) < 0-18-
abcd 5 (2) a 4 abcd 5 a = 1 b 3 bcd 5 b = 1 c 2 cd 5 c = 1, c = 2 c = 1 de = 3 + d + e (d 1) (e 1) = 4 (d 1, e 1) = (1, 4), (2, 2) (d, e) = (2, 5), (3, 3) c = 2 2de = 4 + d + e (2d 1) (2e 1) = 9 (2d 1, 2e 1) = (3, 3) (d, e) = (2, 2) (a, b, c, d, e) = (1, 1, 1, 2, 5), (1, 1, 1, 3, 3), (1, 1, 2, 2, 2) 27 n x 2 y 2 = n x, y n 4 1983 x 2 y 2 = n (x y) (x + y) = n x y + x + y = 2x -19-
x y, x + y x y, x + y n x y, x + y n n n = 4m x + y = 2m, x y = 2 x = m + 1, y = m 1 n n = 2m + 1 x + y = 2m + 1, x y = 1 x = m + 1, y = m 28 x (x x ) {x} (1) m { } { } 1 2,, m m { 3 m } { n,,, m} (2) a {a}, {2a}, {3a},, {na}, 1983 (1) n { n = mq + r r = 0, 1, 2,, m 1 Q, r { { } n mq + r { r = = m} m m} m (2) a i, j {ia} = {ja} -20-
ia ja = m (m Z) a = m i j {a}, {2a}, {3a},, {na}, 29 a, b, c p = a 2 + b 2 + c 2 (1),(2) (1) a, b, c 3 p 3 (2) a, b, c, p a, b, c 3 1983 (1) (3m ± 1) 2 = 3M + 1 a, b, c p = a 2 + b 2 + c 2 a, b, c 3 p = 3A + 1 + 3B + 1 + 3C + 1 = 3 (A + B + C + 1) p 3 (2) a, b, c p p = 3 a 2 + b 2 + c 2 = 3 a = b = c = 1 a, b, c 30 n f(n) n (1) f (n) { f ( n 2) + f (n) } = 12 f(n) (2) f(n 2 + n) (3) n, k f(n k ) = f(n k+4 ) 1983-21-
(1) f (n) = r n = 10q + r (0 r 9) r 0 1 2 3 4 5 6 7 8 9 r 2 0 1 4 9 6 5 6 9 4 1 f(n 2 ) + f(n) 0 2 6 12 10 10 12 16 12 10 f (n) { f ( n 2) + f (n) } 0 2 12 36 40 50 72 112 96 90 f(n) = 2 f(n) 12 f (n) = 1, 2, 3, 4, 6 f ( n 2) = 1, 4, 9, 6, 6 f (n) { f ( n 2) + f (n) } = 1 {1 + 1}, 2 {2 + 4}, 3 {3 + 9}, 4 {4 + 6}, 6 {6 + 6} f(n) = 2 (2) n 2 + n = n (n + 1) f(n 2 + n) (3) n k+4 n k = n k ( n 4 1 ) = n k { (n 2) (n 1) (n + 1) (n + 2) + 5n 2 5 } = n k 1 (n 2) (n 1) n (n + 1) (n + 2) +5 (n 1) n (n + 1) }{{}}{{} 5 3 n k+4 n k 10 f ( n k+4) = f ( n k) -22-
31 a, b a b a 1, a 2, a 3 a b = a 1 + 1 a 2 + 1 a 3 p 1, p 2 p 1 = a 1, p 2 = a 1 + 1 xy zw q 1 q 2 q 1 q 2 a 2 x z w y p 2 p 1 (1) q 2 q 1 a 3 p 2 + p 1 p 2 (2) a 3 q 2 + q 1 q 2 (3) a = a 3 p 2 + p 1, b = a 3 q 2 + q 1 1983 (1) p 1 q 1 = a 1 p 1, q 1 a 1 q 1 = 1, p 1 = a 1 p 2 q 2 = a 1 + 1 a 2 p 2 a 2 = (a 2 a 1 + 1) q 2 p 2 a 2 q 2 p 1, q 1 q 2 a 2 (a 2 a 1 + 1) a 2 a 2 q 2 a 2 = q 2 p 2 = a 2 a 1 + 1 p 2 p 1 q 2 q 1 = p 2q 1 p 1 q 2 ) = (a 1 + 1a2 q 2 q 1 a 1 q 2 q 1 = q 2q 1 a 2 = q 2q 1 q 2 = q 1 = 1-23-
(2) (3) a 3p 2 + p 1 p 2 a 3 q 2 + q 1 q 2 = (a 3p 2 + p 1 ) q 2 p 2 (a 3 q 2 + q 1 ) = p 1 q 2 p 2 q 1 = 1 a b = a 1 1 + a 2 + 1 a 3 a 3 = a 1 + a 2 a 3 + 1 = a 1 (a 2 a 3 + 1) + a 3 a 2 a 3 + 1 a = a 1 (a 2 a 3 + 1) + a 3 = a 3 (a 1 a 2 + 1) + a 1 = a 3 p 2 + p 1 b = a 2 a 3 + 1 = a 3 q 2 + q 1 32 1 S {3m + 7n m, n S} k k 1983 {3m + 7} m=1,2,3 = {10, 13, 16, 19, 22, 25, } {3m + 14} m=1,2,3 = {17, 20, 23, 26, 29, } {3m + 21} m=1,2,3 = {24, 27, 30, } 22 22 33 a, b (1) c = a + b, d = a 2 ab + b 2 1 < c2 d 4 (2) a 3 + b 3 a, b 1984-24-
(1) c = a + b d = a 2 ab + b 2 c 2 d = a2 + 2ab + b 2 a 2 ab + b 2 3ab = 1 + a 2 ab + b 2 3 = 1 + a b + a b 1 3 1 + 2 a b a b 1 = 4 1 1 < c2 d (2) a 3 + b 3 = cd = p n (p : ) k, j c = p j, d = p k (1) 1 < p2j p k = p2j k 4 (p, 2j k) = (2, 1), (2, 2), (3, 1) ( i ) (p, 2j k) = (2, 1) a + b = 2 j, a 2 ab + b 2 = 2 2j 1 (a + b) 2 3ab = 2 2j 3ab = 2 2j 1 3ab = 2 2j 1 a, b ( ii ) (p, 2j k) = (2, 2) a + b = 2 j, a 2 ab + b 2 = 2 2j 2 (a + b) 2 3ab = 2 2j 3ab = 2 2j 2 3ab = 3 2 2j 2 ab = 2 2j 2-25-
a = 2 α, b = 2 β (α + β = 2j 2) 1 a = b a = b = 2 j 1 (iii) (p, 2j k) = (3, 1) a + b = 3 j, a 2 ab + b 2 = 3 2j 1 (a + b) 2 3ab = 3 2j 3ab = 3 2j 1 3ab = 3 2j 1 ab = 3 2j 2 a = 3 α, b = 3 β (α + β = 2j 2) 34 a, b, c a 3 + 2b 3 + 4c 3 = 2abc (1) a, b, c (2) a = b = c = 0 1985 (1) a 3 + 2b 3 + 4c 3 = 2abc a 3 = 2 ( abc b 3 2c 3) a = 2A( Z) 8A 3 + 2b 3 + 4c 3 = 4Abc b 3 = 2 ( Abc c 3 2A 3) b = 2B( Z) 8A 3 + 16B 3 + 4c 3 = 8ABc c 3 = 2 ( ABc A 3 2B 3) c = 2C( Z) (2) A 3 + 2B 3 + 4C 3 = 2ABC A, B, C a, b, c n 2 n a, b, c a = b = c = 0-26-
35 43x + 782y = 1 2 < x + 18y < 12 x, y x y 1987 43x + 782y = 1 1 Euclid 782 = 43 18 + 8 43 = 8 5 + 3 8 = 3 2 + 2 3 = 2 1 + 1 782 43 1 = 3 2 1 = 3 (8 3 2) = 8 + 3 3 = 8 + 3 (43 8 5) = 43 3 8 16 = 43 3 (782 43 18) 16 = 43 291 + 782 ( 16) 1 43 291 + 782 ( 16) = 1 2 1 2 43 (x 291) + 782 (y + 16) = 0 { { x 291 = 782n x = 782n + 291 y + 16 = 43n y = 43n 16 2 < x + 18y < 12 2 < 782n + 291 18 (43n + 16) < 12 2 < 8n + 3 < 12 n = 0, n = 1, n = 1 n (x, y) y = 291 1073 491 = 18.1875, = 18.18644, x 16 59 27 = 18.1851-27-
291 16 36 x x [x] (1) a m [ma] a m < [ma] + 1 a (2) a, b 1 a + 1 = 1 m, n [ma] = [nb] b a, b 1992 (1) (2) ma 1 < [ma] ma [ma] a m < [ma] + 1 a 1 a + 1 = 1, a > 0, b > 0 b [ma] = [nb], m, n N [ma] = [nb] = k k a + k b = k (1) k a m < k a + 1 a 1 k b n < k b + 1 b 2 1 + 2 k m + n < k + 1-28-
m + n = k 1, 2 a = k m, b = k n 37 x, y M, N M = 11x + 2y, N = 18x + 5y (1) M 19 N 19 M N 19 (2) M + x N + y 2,6,18 1992 (1) (2) { M = 11x + 2y { 19x = 5M 2N N = 18x + 5y 19y = 18M + 11N M 19 N 19 M, N 19g M = 19gM, N = 19gN { x = g (5M 2N ) y = g ( 18M + 11N ) x, y g = 1 M, N 19 M + x = 12x + 2y = α N + y = 18x + 6y = β { 3α β = 18x 9α + 6β = 18y α, β G G 18x, 18y x, y G 18 α, β 2 G = 2, 6, 18-29-
38 n (1) n 2 2n + 1 (2) n 2 + 2 2n + 1 n 1992 (1) n 2 2n + 1 p (2) n 2 + 2n + 1 = (n + 1) 2 (n + 1) 2 p n n + 1 p n 2 2n + 1 n 2 + 2 = k (2n + 1) n 2 2kn + 2 k = 0 n = k + k 2 2 + k k 2 2 + k = m 2 m ( k + 1 ) 2 9 2 4 = m2 (2k + 1) 2 9 = 4m 2 (2k + 1 2m) (2k + 1 + 2m) = 9 (2k + 1 + 2m, 2k + 1 2m) = (9, 1), (3, 3) (k.m) = (2, 2), (1, 0) n = 4, 1-30-
39 k 0 x 2 y 2 = k (a, b) a, b (1) x 2 y 2 = k k 8 (2) x 2 y 2 = k 1992 (1) x 2 y 2 = k x = 2m + 1, y = 2n + 1 k = (2m + 1) 2 (2n + 1) 2 = 4m (m + 1) 4n (n + 1) m(m + 1), n(n + 1) k 8 (2) k = 8K (x + y) (x y) = 8K x + y = 4K, x y = 2 x = 2K + 1, y = 2K 1 k 8-31-
40 n P n n = 2 a m(a m ) (1) P n k n n = k (2) P 1 (1) P n n N 1 2N (N + 1) 1993 (1) k k = 2 0 k k k = 2K K = 2 a m(a m ) k = 2 a+1 m (2) (1, 2), (2, 4), (3, 6),, (N, 2N) N 2N + 1 (k, 2k) Pigeonhole Principle Web If more than half of the integers from 1, 2,..., 2n are selected, then some two of the selected integers are mutually prime. -32-
41 5, 17, 257 2 2 + 1, 2 4 + 1, 2 8 + 1 n 2 n + 1 n 2 (1) a x a + 1 (x + 1)f(x) f(x) (2) 2 m a m a + 1 (3) 1994 (1) (2) x a + 1 = 1 ( x) a = (1 + x) (1 x + x 2 x 3 + + ( x) a 1) f (x) = 1 x + x 2 x 3 + + ( x) a 1 m a + 1 = (1 + m) (1 m + m 2 + + ( m) a 1) 1 + m 3, 1 m + m 2 + + ( m) a 1 1 m + m 2 3 m a + 1 (3) a 3 2 a + 1 2 a + 1 a = 2, 1 a a a = 2 k b(k 1, b : odd) k, b b \= 1 ( 2 a + 1 = 2 2k) b + 1 b = 1 2 a + 1 a = 2 k (k = 0, 1, 2, ) -33-
42 a, b, c, d a 2 + b 2 + c 2 = d 2 (1) d 3 a, b, c 3 (2) d a, b, c 1994 (1) A 3 r A r a 0 a 2 0 a ±1 a 2 1 a 2, b 2, c 2 {0, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1} 4 d 2 3 {0, 0, 1}, {0, 1, 1} {0, 1, 1} d 2 2 {0, 0, 1} 2 3 (2) a = 2m a 2 = 4m 2 a = 2m + 1 a 2 = 4 ( m 2 + m ) + 1 4 0 1 a 2, b 2, c 2 4 {0, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1} 4 d 2 4 {0, 0, 1} a, b, c 2 2 a, b, c -34-
(1) α, β 43 [ 1 ] αx βy = 0 [ 2 ] α β = a 1 + a 2 + 1 a 3 + 1 a 4 1 (a 1, a 2, a 3, a 4 ) 1 a 1 + a 2 + 1 a 3 a 1 a 2 a 3 + a 1 + a 3 p, a 2 a 3 + 1 q αq βp (2) 157x 68y = 3 1993 (1) [ ] αx = βy x, y α, β x = βm, y = αm (m Z) [ ] a 1 + 1 a 3 a 2 + 1 = a 1 + a 3 a 2 a 3 + 1 = a 1a 2 a 3 + a 1 + a 3 a 2 a 3 + 1 p = a 1 a 2 a 3 + a 1 + a 3, q = a 2 a 3 + 1 a, b a, b, c ac + b a ac + b a g b g a, b c + b a = ac + b a -35-
1 a 3 a 3 + 1 a 4 1 a 2 + a 3 + 1 a 4 1 a 1 + 1 a 2 + a 3 + 1 a 4 α β = a 1 1 + a 2 + 1 a 3 + 1 a 4 1 = a 1 + a 2a 3a 4+a 2+a 4 a 3a 4+1 a 3 a 4 + 1 = a 1 + a 2 a 3 a 4 + a 2 + a 4 = a 1a 2 a 3 a 4 + a 1 a 2 + a 1 a 4 + a 3 a 4 + 1 a 2 a 3 a 4 + a 2 + a 4 = (p a 1 a 3 ) a 4 + a 1 a 2 + a 1 a 4 + a 3 a 4 + 1 (q 1) a 4 + a 2 + a 4 = pa 4 + a 1 a 2 + 1 qa 4 + a 2 α = pa 4 + a 1 a 2 + 1, β = qa 4 + a 2 αq βp = (pa 4 + a 1 a 2 + 1) q (qa 4 + a 2 ) p = qa 1 a 2 + q pa 2 = a 1 a 2 (a 2 a 3 + 1) + a 2 a 3 + 1 (a 1 a 2 a 3 + a 1 + a 3 ) a 2 = 1 (2) 157 68 = 2 + 21 68 = 2 + 1 1 = 2 + 68 3 + 5 21 21 1 = 2 + 3 + 1 = 2 + 21 5 1 3 + 1 4 + 1 5-36-
p, q p = 2 3 4 + 2 + 4 = 30 q = 3 4 + 1 = 13 157 13 68 30 = 1 157x 68y = 3 1 157 39 68 90 = 3 2 1 2 157 (x 39) 68 (y 90) = 0 157 68 x 39 = 68m, y 90 = 157m x = 39 + 68m, y = 90 + 157m (m Z) 44 n (1) n n (2) (n 1)n(n + 1) n 1995 (1) n a = nq +1 n g a nq g a nq = 1 g 1 g = 1 a n (2) n 1 nq + 1 n nq + 1 (n 1)(n + 1) (n 1) (n + 1) = A (nq + 1), A 1 (n 1) (n + 1) nq + 1 n 2 nq 2 0 n 2 4 2q 2 0 1 q q = 0, 1, 1 n + 1-37-
45 n f(n) (A) f(1) = 1 ( (B) p, a f(p a ) = p a 1 1 ) p (C) m, n f(mn) = f(m)f(n) (1) n(n 2) n = p a 1 1 p a 2 2 pa r r (a i p 1, p 2,, p r 1) f(n) n (2) f(n) = 1 3 n n = 2a 3 b (a 1, b 1) 1993 (1) p i a i, p j a j (2) f (n) = f (p 1 a1 p 2 a2 p r a r ) a = f (p 1 a 1 ) f (p 2 a 2 ) f (p r r ) ( a = p 1 1 1 1 ) ( a p 2 2 1 1 ) ( a p r r 1 1 ) p 1 p 2 p r ) ) ) = n (1 (1 1p1 1p2 (1 1pr f (n) ) ) ) (1 n = (1 1p1 1p2 (1 1pr f (n) = 1 3 n ( 1 1 ) ( 1 1 ) ) (1 1pr = 1 p 1 p 2 3 3 (p 1 1) (p 2 1) (p 3 1) (p r 1) = p 1 p 2 p 3 p r p 1 < p 2 < p 3 < < p r p 1 \= 2 p 1, p 2, p 3. p r p 1 = 2 p 2 = 3 p 3 (p 3 1) (p r 1) = p 3 p r -38-
p 3 n = 2 a 3 b (a 1, b 1) 46 p, q (1) (p, 0) (0, q) x y (2) 0 < x < p, 0 < y < q, x p + y q < 1 (x, y) p, q 1989 (1) x p + y q = 1 (0 < x < p, 0 < y < q) 1 x, y qx + py = pq qx = p (q y) p, q x p 0 < x < p p 1 (2) x p + y q < 1 (0 < x < p, 0 < y < q) S 0 < x < p, 0 < y < q 2S (p 1) (q 1) S = 1 (p 1) (q 1) 2-39-
x = k(k = 1, 2,, p 1) 0 < y < q qk p [ q qk ] p p 1 [ q qk ] [ = q q p p k=1 [ ] [ (p 1) q (p 2) q = + p p ] [ + q 2q p [ ] [ ] rq (p 1) q + 1 = q p p [ ] [ ] rq (p 1) q + = q 1 p p ] + + ] [ + + q [ ] q p S [ ] [ ] (p 1) q (p 2) q S + S = + + + p p [ ] [ ] [ ] q 2q (p 1) q + + + + p p p = (q 1) (p 1) S = 1 (q 1) (p 1) 2 [ ] rq p [ ] q p ] (p 1) q p rq p (p r)q p 0 rq rq q p [ ] rq + 1 p [ ] (p r)q p 47 p n 2n = 65536 n 1989-40-
65536 = 2 16 = 4 8 n 2n = 65536 n n = 4 48 (1) a a 2 (2) a 2 2b 2 a, b (3) a 2 2b 2 + 3c 2 6d 2 = 0 a, b, c, d a = b = c = d = 0 1991 (1) a a = 3m ± 1 a 2 = (3m ± 1) 2 = 3(3m 2 ± 2m) + 1 (2) a 3 R(a) a, b a 2 2b 2 a 0 1 1 b 1 0 1 R(a 2 ) 0 1 1 R(b 2 ) 1 0 1 R(a 2 2b 2 ) 1 1 2 a 2 2b 2 a, b (3) a 2 2b 2 = 3 ( 2d 2 c 2) a, b a = 3A, b = 3B 9A 2 18B 2 = 3 ( 2d 2 c 2) c 2 2d 2 = 3 ( 2B 2 A 2) c, d c = 3C, d = 3D A 2 2B 2 + 3C 2 6D 2 = 0 a, b, c, d 3 a b = c = d = 0-41-
49 0 < a b 1 a, b f(n) = an 3 + bn n f(n) n f(n) a, b 1991 f (1) = a + b Z 1 f (3) = 27a + 3b Z 2 f (2) = 8a + 2b = even 3 4a + b Z 4 1 4 3a 0 < a b 1 a = 1 3, 1 b = 2 3, 1 a = 1 3, b = 2 3 f (n) = 1 3 n3 + 2 (n 1) n (n + 1) + 3n n = 3 3 a = b = 1 50 a, b u, v u + v 3 x 2 + ax + b = 0 u, v 3 1999 ( u + v ) 2 ( 3 + a u + v ) 3 + b = 0 u 2 + 3v 2 + au + b + (2uv + av) 3 = 0 2uv + av \= 0 3 = u 2 + 3v 2 + au + b 2uv + av { 2uv + av = 0 v = 0, 2u + a = 0 u 2 + 3v 2 + au + b = 0-42-
( i ) v = u = n m, n m 1 m ( n ) 2 ( n + a + b = 0 m m) n 2 + amn + bm 2 = 0 n 2 = m (an + bm) m n 2 m = 1 u ( ii ) 2u + a = 0 u 2 + 3v 2 + au + b = 0 1 4 a2 + 3v 2 1 2 a2 + b = 0 12v 2 a 2 + 4b = 0 v = p q p, q q 1 ( ) 2 p 12 a 2 + 4b = 0 q 12p 2 a 2 q 2 + 4bq 2 = 0 12p 2 = q 2 ( a 2 4b ) p, q q 2 12 q = 1, 2 q = 1 v = p 12p 2 a 2 + 4b = 0 a 2 = 4 ( 3p 2 + b ) a 2u = a u q = 2 p, q p 12p 2 4a 2 + 16b = 0 3p 2 a 2 + 4b = 0 a a = 2A + 1, p = 2P + 1 3(2P + 1) 2 (2A + 1) 2 + 4b = 0 12P 2 + 12P + 2 4A 2 4A + 4b = 0 6P (P + 1) 2A(A + 1) = (2b + 1) -43-
u, v v = 0 v \= 0 u + v 3 u v 3 { 2u = a u 2 3v 2 = b u a 2 12v 2 = 4b a 2 = 4 ( 3v 2 b ) a u a = 2A A 2 = 3v 2 b v = r r, s s 1 s A 2 + b 3 = r s 2 2 s = 1 A 2 + b u, v 51 (1) (x 2 ny 2 )(z 2 nt 2 ) = (xz + nty) 2 n(xt + yz) 2 (2) x 2 2y 2 = 1 (x, y) x > 100 1998 (1) ( x 2 ny 2) ( z 2 nt 2) = ( x + ny ) ( x ny ) ( z + nt ) ( z nt ) = {( x + ny ) ( z + nt )} {( x ny ) ( z nt )} = { xz + nyt + (xt + yz) n } { xz + nyt (xt + yz) n } = (xz + nyt) 2 n(xt + yz) 2-44-
(2) (1) n = 2, z = 3, t = 2 x 2 2y 2 = (3x + 4y) 2 2(2x + 3y) 2 { xn+1 = 3x n + 4y n y n+1 = 2x n + 3y n x 1 = 1, y 1 = 1 x 2 1 2y 2 1 = 1 x 2 k 2y 2 k = 1 x 2 k+1 2y 2 k+1 = 1 n (x n, y n ) x 2 2y 2 = 1 (x 2, y 2 ) = (7, 5) (x 3, y 3 ) = (41, 29) (x 4, y 4 ) = (259, 169) x > 100 (259, 169) { ax + y = 2 52 x (a + 1)y < 2a x, y a 1997 a = 2 y x a ( 2 y x x ) + 1 y < 2 2 y x x 2 2y + y 2 xy < 4 2y x 2 xy + y 2 < 4 ( x y ) 2 3y 2 + 2 4 < 4 3y 2 4 < 4 y2 < 16 3 y = 1, 2-45-
y = 1 x 2 x < 3 x = 1, 2 y = 2 x 2 2x < 0 x = 1 53 n(n 1) n f(n) = a n = i f(n) i 2 a n+k = a n n k 2001 (n + k) (n + k 1) f (n + k) f (n) = 2 (2k 2) n + k (k 1) = 2 = 1 (k 1) (2n + 1) 2 n (n 1) 2 a n+k = a n i f(n+k) f(n) = 1 f (n + k) f (n) = 4m 1 k (2n + k 1) = 4m 2 k (2n + k 1) = 8m k 8 k + 2n + k 1 = 2 (n + k) 1 = k 2n + k 1 k k = 2K + 1 2n + 2K + 1 1 = 8M n + k = 4M n k -46-
54 p, q f(x) = x 2 + px + q (1) a f(x) = 0 a (2) f(1) f(2) f(x) = 0 2000 (1) (2) α = u v u, v v 1 ( u ) 2 ( u ) + p + q = 0 v v u 2 = v (pu + qv) v u u, v v = 1 α f (n) = n 2 + pn + q f (n + 1) f (n) = 2n + 1 + p f (n + 2) f (n + 1) f (n + 1) + f (n) = 2 f (n + 2) = 2f (n + 1) f (n) + 2 ( i ) f(1), f(2) ( ii ) f(k), f(k + 1) f (k + 2) = 2f (k + 1) f (k) + 2 n 1 f(n) f (k) = 2f (k + 1) f (k + 2) + 2 k, k 1 f(k), f(k 1) f(k 2) n f(n) f(x) = 0-47-
55 p a, b, c a + b + c, a 2 + b 2 + c 2, a 3 + b 3 + c 3 p (1) ab + bc + ca p (2) abc p (3) a, b, c p 1986 (1) 2 (ab + bc + ca) = (a + b + c) 2 ( a 2 + b 2 + c 2) 2 (ab + bc + ca) p(> 3) p ab + bc + ca p (2) a 3 + b 3 + c 3 3abc = (a + b + c) ( a 2 + b 2 + c 2 ab bc ca ) 3abc = a 3 + b 3 + c 3 (a + b + c) ( a 2 + b 2 + c 2 ab bc ca ) 3abc p p 3 abc p (3) a + b + c = pl ab + bc + ca = pm abc = pn a, b, c x 3 plx 2 + pmx pn = 0 a 3 = p ( la 2 ma + n ) a p b, c -48-
56 n (1) n m n m (2) n m n m (3) n m m n 2007 (1) n = 3p + 1 n m = (3p + 1) m = 1 + m C 1 (3p) + m C 2 (3p) 2 + = 1 + 3M n m 3 (2) n = 3p 1 m n m = (3p 1) m = 1 + m C 1 (3p) m C 2 (3p) 2 + = 1 + 3M n m 3 (3) n = 3p + r(r = 0, 1, 1) n m = (3p + r) m = r m + m C 1 r m 1 (3p) + m C 2 r m 1 (3p) 2 + = r m + 3M 1 (: r = 1) = ( 1) m (: r = 1) 0 (: r = 0) n m 3 2 r = 1, m n -49-
57 p n 0 (1) m 0 m n 1 p n+1 p m p m+1 (2) 1 p n+1 x, y xy p n+1 (x, y) 2007 (1) [ p n+1 p m ] [ ] p n+1 p m+1 = p n+1 m p n m = p n m (p 1) (2) x, y x = p i q, y = p j r (1 i, j n + 1) i + j n + 1 (i, j) p i p i+1 x p j p j+1 y p n+1 2p n+1 1 p n i (p 1) p n j (p 1) + 2p n+1 1 0 i,j n,i+j n+1 = n n i=0 j=n+1 i = (p 1) 2 n p n i (p 1) p n j (p 1) + 2p n+1 1 n i=1 j=n+1 i p 2n i j + 2p n+1 1 n = (p 1) 2 p n i pi 1 p 1 + 2pn+1 1 i=0 n ( = (p 1) p n p n i) + 2p n+1 1 = (p 1) i=0 { } p n (n + 1) pn+1 1 + 2p n+1 1 p 1 = (p 1) p n (n + 1) p n+1 + 1 + 2p n+1 1 = (n + 2) p n+1 (n + 1) p n -50-
58 p a, b, c, d a + b + c + d = 0, ad bc + p = 0, a b c d a, b, c, d p 2007 d = a b c ad bc + p = a ( a b c) bc + p = 0 p = a 2 + ab + ac + bc p = (a + b) (a + c) a + b c + d, a + b = (c + d) a + b 0 c + d a + b = p, a + c = 1 b = p a, c = 1 a, d = a p 1 a b, c d a p a, 1 a a p 1 p 2a p + 2 p 2a = p + 1 a = p + 1 2, b = p 1 (1) m 2, c = p + 1 2 (2) a, b f(x), d = p 1 2 59 pm m 2 m 1 f(x) = x 4 + ax 2 + bx a 2 m a, b p f(m) m 2 m 1 2005-51-
(1) m = [p] + 3 pm m 2 m 1 = p m 1 1 = p m [p] + 3 1 1 p < p + 1 1 < 1 [p]+2 [p]+3 pm m 2 m + 1 = 0 m p = 0 (2) f (m) m 2 m + 1 = (a + b + 3) m m2 + m + a + 2 + m 2 m + 1 m = 0 a + 2 a m = 1 b + 1 b f (m) m 2 m 1 m a + b + 3 = 0 60 n 2 (1) n 3 n 6 (2) n 5 n 30 2005 (1) n 3 n = (n 1)n(n + 1) 3! = 6-52-
(2) (n 2) (n 1) n (n + 1) (n + 2) = n ( n 2 1 ) ( n 2 4 ) = n 5 5n 3 + 4n 5! = 120 n 5 n = (n 2) (n 1) n (n + 1) (n + 2) + 5n 3 5n = (n 2) (n 1) n (n + 1) (n + 2) + 5 ( n 3 n ) n 5 n 61 3 9999 a a 2 a 10000 2005 a (a 1) = 10000Q = 2 4 5 4 Q a a 1 a = 5 4 M (M : ), a 1 = 2 4 N a = M (M :), a 1 = 2 4 5 4 N a = 10000N + 1 > 10001 a = 5 4 M, a 1 = 2 4 N a 5 4 M 1 = 2 4 N 625M + 16N = 1 625 = 16 39 + 1 625 1 + 16 ( 39) = 1 625 (M 1) + 16 (N + 39) = 0 625 16 N + 39 = 625m N = 625m + 39 a = 2 4 (625m + 39) + 1 = 10000m + 625-53-
a < 10000 a = 625 ax + by = c (x, y) c 73x + 18y = 3 1 73 18 73 = 18 4 + 1 73 3 + 18 ( 12) = 3 2 1 2 73(x 3) + 18(y + 12) = 0 73(x 3) = 18(y + 12) 73 18 x 3 = 18m, y + 12 = 73m x = 3 18m, y = 73m 12 62 (1) p, q 2 log p q (2) a, b 2 log a b a a b b a (3) a, b 2 log a b a, b c, α, β a = c α, b = c β 2005 (1) log p q = m n (m, n n 1) p m n = q p m = q n p m q p, q (2) log a b = m n (m, n n 1) -54-
a m n = b a m = b n a b a b b a (3) a, b, p a = p K b = p L Km = Ln m, n K n L m K = nn, L = mn N a = p nn = ( p N )n b = p mn = ( p N )m 63 (1) p, 2p + 1, 4p + 1 p (2) q, 2q + 1, 4q 1, 6q 1, 8q + 1 q 2005 p 3m 1 3m 3m + 1 2p + 1 6m 1 6m + 1 6m + 3 4p + 1 12m 3 12m + 1 12m + 5 p, 2p + 1, 4p + 1 ( i ) 12m 3 = 3 (4m 1) 4m 1 = 1 2m = 1 ( ii ) 3m m = 1 p = 3, 2p + 1 = 7, 4p + 1 = 13-55-
(iii) 6m + 3 = 3(2m + 1) m = 0 p = 1 p = 3 p (2p + 1) (4p + 1) = 8p 3 + 6p 2 + p = 9p 3 + 6p 2 p 3 + p = 9p 3 + 6p 2 (p 1)p(p + 1) p, 2p + 1, 4q + 1 p = 3 ( i ) q = 5m q m = 1 2q + 1 = 10m + 1 = 11 4q 1 = 20m 1 = 19 6q 1 = 30m 1 = 29 8q + 1 = 40m + 1 = 41 ( ii ) q = 5m + 1 2q + 1 = 10m + 3 4q 1 = 20m + 3 6q 1 = 30m + 5 = 5 (6m + 1) 8q + 1 = 40m + 6 = 2 (20m + 3) 5 (6m + 1) m = 0 q = 1 (iii) q = 5m + 2 2q + 1 = 10m + 5 = 5 (2m + 1) m = 0 q = 2, 2q + 1 = 5, 4q 1 = 7, 6q 1 = 11, 8q + 1 = 17 (iv) q = 5m + 3 8q + 1 = 40m + 25 = 5 (20m + 3) ( v ) q = 5m + 4 4q 1 = 20m + 15 = 5 (4m + 3) q = 2 q = 5 q (2q + 1) (4q 1) (6q 1) (8q + 1) = 384q 5 + 80q 4 60q 3 + q = 385q 5 + 80q 4 60q 3 q 5 + q -56-
q, 2q + 1, 4q 1, 6q 1, 8q 1 q = 2, q = 5 64 xy (x, y) x, y 1 x 100, 1 y 100 x + y 2 2x + 3y 5 2005 2x + 3y = 5m 2 (x m) = 3 (y m) { x m = 3t y m = 2t x + y = 2m t t = 2k { x = m 6k y = m + 4k m, k 1 m 6k 100, 1 m + 4k 100 1 + 6k m 100 + 6k, 1 4k m 100 4k m 1 + 6k 100 4k, 1 4k 100 + 6k 99 10 k 99 10 ( i ) 9 k < 0 1 4k m 100 + 6k -57-
m 100 + 6k (1 4k) + 1 = 100 + 10k k 1 k= 9 (100 + 10k) = (10 + 90) 9 1 2 = 450 ( ii ) k = 0 1 m 100 100 (iii) 1 k 9 1 + 6k m 100 4k m 100 4k (1 + 6k) + 1 = 100 10k 9 (100 10k) = (10 + 90) 9 1 2 = 450 k=1 1000 65 a 3 b 3 = 217 (a, b) 2005 a 3 b 3 = 217 (a b) ( a 2 + ab + b 2) = 7 31 a 2 + ab + b 2 > 0, a b > 0 a 2 + ab + b 2 > a b { a b = 1 a 2 + ab + b 2 = 217 { a b = 217 1 a 2 + ab + b 2 = 1 { a b = 7 2 a 2 + ab + b 2 = 31 { a b = 31 3 a 2 + ab + b 2 = 7 4-58-
( i ) 1 (b + 1) 2 + b (b + 1) + b 2 = 217 3b 2 + 3b 216 = 0 b 2 + b 72 = 0 (b + 9) (b 8) = 0 (a, b) = ( 8, 9), (9, 8) ( ii ) 2 (b + 217) 2 + b (b + 217) + b 2 = 1 3b 2 + 3 217b + 216 218 = 0 b 2 + 217b + 72 218 = 0 D = 217 2 4 72 218 < 0 (iii) (iv) 3 (b + 7) 2 + b (b + 7) + b 2 = 31 3b 2 + 3 7b + 18 = 0 b 2 + 7b + 6 = 0 (b + 1) (b + 6) = 0 (a, b) = (6, 1), (1, 6) 4 (b + 31) 2 + b (b + 31) + b 2 = 7 3b 2 + 3 31b + 954 = 0 b 2 + 31b + 318 = 0 D = 31 2 4 318 < 0 66 n A(n) n 2 + 3n + 2 (1) j + k = 7 j k A(j) = A(k) (2) j + k = 97 j k A(j) = A(k) (3) j 5 n A(n + j) = A(n) 2005-59-
(1) f (n) = n 2 + 3n + 2 k = 7 j f (j) f (k) = f (j) f (7 j) = j 2 + 3j + 2 (7 j) 2 3 (7 j) 2 = 70 + 20j = 10 (2j 7) A (j) = A (k) (2) k = 97 j f (j) f (k) = f (j) f (97 j) = j 2 + 3j + 2 (97 j) 2 3 (97 j) 2 = 97 2 3 97 + 3j 2 + 2 97j = 100 97 + 100 2j A (j) = A (k) (3) f (n + j) f (n) = = (n + j) 2 + 3 (n + j) + 2 n 2 3n 2 = 2nj + j 2 + 3j = 2j (n + j + 3) i A (j) = A (k) 67 (1) 25m + 17n = 1623 (m, n) 1 (2) 25m + 17n = 1623 (m, n) 2005 (1) 1623 = 25 64 + 23 23 = 17 1 + 6-60-
25m + 17n = 1623 25 (m 64) + 17n = 23 25 (m 64) + 17 (n 1) = 6 1 25 = 17 + 8 17 = 8 2 + 1 1 = 17 8 2 = 17 (25 17) 2 = 25 ( 2) + 17 3 25 ( 12) + 17 16 = 6 1 m 64 = 12, n 1 = 16 m = 52, n = 17 (2) 25 (m 52) + 17 (n 17) = 0 25 17 m 52 = 17k, n 17 = 25k m = 52 + 17k, n = 17 25k (k Z) -61-
68 a b 1 a b n p M p (n) p 1 x, y x y = M p (xy) xy M 11 (6 (1) ) = 2 (2) (3) (4) (5) x, y, z p 1 a p a p 1 p 1 M p ( (6) ) = (7) M p p x, y p 1 M(ax) = M(ay) a(x y) (8) (9) (10) (11) x = y (12) (13) (14) (15) (16) (17) M(1a), M(2a),, M((p 1)a) M(1a) M(2a) M((p 1)a) = 1 2 (18) (19) M M(1a) M(2a) M((p 1)a) = M( (20) ) 1 2 (21) (22) x y = y x = (23) M( (6) ) = (7) (1) 1 (2) 2 (3) 3 (4) 4 (5) 0 (6) a (7) a p 1 (8) a p (9) a p+1 (10) x y (11) x y (12) xy (13) x + y (14) x \= y (15) M(ax) = M(ay) (16) x = y (17) p + 1 (18) p (19) p 1 (20) M(ax) \= M(ay) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) p 1 (31) x y = 0 (32) x y = y x (33) x y z = y z x = z x y (34) x (y z) = (x y) z (35) x (y + z) = x y + x z 2005-62-
(1) 4 (2)(3) 32 (4)(5) 34 (6) 7 (7) 1 (8)(9) 18 (10)(11) 26 (12)(13) 22 (14)(15) 14 (16)(17) 20 (18)(19) 19 (20) 7 (21)(22) 19 (23) 1 n p M p (n) p 1 x, y x y = M p (xy) xy M 11 (6 4 ) = 2 x y = y x x (y z) = (x y) z x, y, z p 1 a p a p 1 p 1 M p (a p 1 = 1 M p p x, y p 1 M(ax) = M(ay) a(x y) p x = y x \= y M(ax) \= M(ay) M(1a), M(2a),, M((p 1)a) M(1a) M(2a) M((p 1)a) = 1 2 p-1 M M(1a) M(2a) M((p 1)a) = M( a p 1 ) 1 2 p-1 x y = y x = 1 M p (a p 1 ) = 1-63-
69 1 2 2 n P (n) { n P (n) = 1, n P (n) n P (2) = 1, P (3) = 1, P (4) = 2, P (5) = 1, P (6) = 2 (1) P (a) = 5 2 a a = (2) n 2 n 100 P (n) P (b) = b 2 b 100 (3) c 2 n 2 n c P (n) 11 c, 2005 25 7 3 121 168 (1) a 5 5 a = 5 5 = 25 (2) 5 2 = 25, 7 2 = 49, 11 2 = 121 n 2 n 100 P (n) 7 P (b) = 7 b 2 b 100 7 7 = 49, 7 11 = 77, 7 13 = 91 (3) 11 11, 13 13 = 169 121 169-1=168-64-
70 1 x + 1 2y + 1 3z = 4 1 3 (x, y, z) (1) x = 1 y, z (2) x (3) 1 2005 (1) x = 1 1 + 1 2y + 1 3z = 4 3 1 2y + 1 3z = 1 3 3z + 2y = 2yz 2y (z 1) 3 (z 1) = 3 (2y 3) (z 1) = 3 (2y 3, z 1) = (1, 3), (3, 1) (y, z) = (2, 4), (3, 2) (2) 1 2y 1 2, 1 3z 1 3 4 3 = 1 x + 1 2y + 1 3z 1 x + 1 2 + 1 3 1 2 1 x x 2 x = 1, 2-65-
(3) x = 2 1 2 + 1 2y + 1 3z = 4 3 1 2y + 1 3z = 5 6 3z + 2y = 5yz 25yz 10y 15z = 0 5y (5z 2) 3 (5z 2) = 6 (5y 3) (5z 2) = 6 (5y 3, 5z 2) = (1, 6), (6, 1), ( 1, 6), ( 6, 1) (5y, 5z) = (4, 8), (9, 3), (2, 4), ( 3, 1) y, z (x, y, z) = (1, 2, 4), (1, 3, 2) 71 n n n 1 f(n) n = 12 1, 5, 7, 11 4 f(12) = 4 f(1) = 1, p f(p) = p 1 (1) f(77) (2) f(pq) = 24 2 p, q p < q (3) k, n f(2 k 3 n ) k n 2005 (1) 1 77 7 11 11 7 77 f (77) = 77 11 7 + 1 = 60 (2) p, q p < q f (pq) = pq p q + 1 = (p 1) (q 1) = 24 (p 1, q 1) = (1, 24), (2, 12), (3, 8), (4, 6) (p, q) = (3, 13), (5, 7) (3) 1 2 k 3 n 2 2 k 1 3 n 3 2 k 3 n 1 6 2 k 1 3 n 1 f ( 2 k 3 n) = 2 k 3 n 2 k 1 3 n 2 k 3 n 1 + 2 k 1 3 n 1 = 2 k 1 3 n 1 (6 3 2 + 1) = 2 k 3 n 1-66-
ϕ (n) f 1 n, 2 n, 3 n, n n ϕ (n) ( i ) p ϕ ( p k) ( = p k p k 1 = p k 1 1 ) p ( ii ) a, b (iii) a ϕ (ab) = ϕ (a) ϕ (b) a = p 1 α1 p 22 p 3 α3 p m α m ) ) ) ( ϕ (a) = a (1 (1 1p1 (1 1p2 1p3 1 1 ) p m 72 n x, y, z x n + y n + z n = xyz 1 (1) n = 1 1 (x, y, z) x y z (2) n = 3 1 (x, y, z) 2006 (1) { x + y + z = xyz 0 < x y z -67-
1 = 1 xy + 1 yz + 1 zx 1 x 2 3 x 2 3 x = 1 1 + y + z = yz (y 1) (z 1) = 2 (y 1, z 1) = (1, 2) y = 2, z = 3 (2) x 3 + y 3 + z 3 = xyz x, y, z x 3 + y 3 + z 3 3 x 3 3 y 3 z 3 = xyz xyz 3 xyz xyz 0 x 3 + y 3 + z 3 = xyz x, y, z (x, y, z) (A) x, y, z x 2 + y 2 + z 2 = xyz x y z (1) (A) (x, y, z) y 3 (2) (a, b, c) (A) (b, c, z) (A) z (3) (A) (x, y, z) 2006-68-
(1) x 2 + y 2 + z 2 = xyz z 2 xyz + ( x 2 + y 2) = 0 z D = (xy) 2 4 ( x 2 + y 2) 0 1 4 x 2 + 4 y 2 4 y 2 + 4 ( x y) y2 y 2 8 y 3 y 3 y = 3 D = (3x) 2 4 ( x 2 + 3 2) 0 5x 2 36 x 3 x x = 3 9 + 9 + z 2 = 9z (z 3) (z 6) = 0 z = 3, z = 6 (x, y, z) = (3, 3, 3), (3, 3, 6) (2) a 2 + b 2 + c 2 = abc, a b c z 2 bcz + b 2 + c 2 = 0 D = b 2 c 2 4 ( b 2 + c 2) = b 2 c 2 4 ( abc a 2) = (bc 2a) 2 z = bc ± (bc 2a) 2 = bc a, a bc a c = c (b 1) a 3c a > 0 b 2 + c 2 + z 2 = zbc, b c z z (3) { a1 = 3, b 1 = 3, c 1 = 3 a n+1 = b n, b n+1 = c n, c n+1 = b n c n a n -69-
{(a n, b n, c n )} (n=1,2,3, ) a 2 n + b 2 n + c 2 n = a n b n c n, a n b n c n c n+1 = b n c n a n c n + 2c n a n > c n (A) (x, y, z) 73 a, b, c 3a = b 3, 5a = c 2 d 6 a d d = 1 (1) a 3 5 (2) a 3 5 (3) a 2006 (1) 3a = b 3, 5a = c 2 b 3 3 b 3 c 2 5 c 5 b = 3b, c = 5c a = 3 2 b 3 = 5c 2 1 a 3 5 (2) (3) 1 b = 5b, c = 3c a = 3 2 5 3 b 3 = 5 3 2 c 2 a = 3 2 5 3 A -70-
3 3 5 3 A = b 3, 3 2 5 4 A = c 2 A A = d 6 a = 3 2 5 3 d 6 d = 1 a = 3 2 5 3 = 1125 (2) (3) (2) a 3,5 p l b p m c p n l = 3m = 2n l = 6k a = Ap 6k = A ( p k) 6 p k = 1 a 3,5 (3) a = 3 i 5 j 3 i+1 5 j = b 3, 3 i 5 j+1 = c 2 i + 1 = 3m, i = 2n 1 j = 3m, j + 1 = 2n 2 1 3m 2n = 1 3 (m 1) 2 (n 1) = 0 m 1 = 2k, n 1 = 3k i = 3 (2k + 1) 1 = 6k + 2 3 j = 6k + 3 a = 2 2 5 3( 2 k 5 k ) 6-71-
2 k 5 k = 1 a = 3 2 5 3 = 1125 74 (a), (b) p, 2 q, r (a) p, q, r 2 (b) p + q + r = 132 (1) q, r (2) p, q, r 2006 (1) n ( i ) n = 2N n 2 = 4N 2 ( ii ) n = 2N + 1 n 2 = 4(N 2 + N) + 1 ( ) q, r q 2 + r 2 = (2Q + 1) 2 + (2R + 1) 2 = 4 ( Q 2 + R 2 + Q + R ) + 2 p ( ) q, r (2) q, r q r p, q, r q, r r r = 2 p 2 = q 2 + 4 (p q) (p + q) = 4 p q = 1, p + q = 4 2p = 5, 2q = 3-72-
p, q p 2 = q 2 + r 2 (p r) (p + r) = q 2 1 p r < p + r p + r = q 2, p r = 1 p = 1 ( q 2 + 1 ), r = 1 ( q 2 1 ) 2 2 p + q + r = 132 1 ( q 2 + 1 ) + 1 ( q 2 1 ) + q = 132 2 2 q 2 + q 132 = 0 (q 11) (q + 12) = 0 q q = 11 r = 1 2 (112 1) = 60, p = 60 + 1 = 61 p 2 = q 2 + r 2 p, q, r p = m 2 + n 2, q = m 2 n 2, r = 2mn r m = n = 1 q = 0 p, q q = (m + n)(m n) m n = 1 m = n + 1 p = (n + 1) 2 + n 2, q = (n + 1) 2 n 2, r = 2n(n + 1) p + q + r = 132 2 (n + 1) 2 + 2n(n + 1) = 132 2n 2 + 3n 65 = 0 (n 5) (2n + 13) = 0 n = 5 75 2 n n n 2 + 2 n = 3 2006 n ( n 2 + 2 ) = n ( n 2 1 ) + 3n = (n 1) n (n + 1) + 3n -73-
3 n 2 + 2 8 n = 3 76 1 (1) k k k + 1 (2) n 3 n 2 k n 2 = 2k + 1 n, k, k + 1 (3) 2006 (1) k, k + 1 g g k + 1 k = 1 g = 1 (2) n = 2m + 1 n 2 = 2 ( 2m 2 + 2m ) + 1 k = 2m 2 + 2m k + 1 = 2m 2 + 2m + 1 k, k + 1 (1) ( 2m 2 + 2m ) m (2m + 1) = m 1 (2m + 1) 2 (m) = 1 2 k = 2m 2 + 2m n = 2m + 1 g 1 1 g 1 m 2 g 1 1 g 1 = 1 ( 2m 2 + 2m + 1 ) m (2m + 1) = m + 1 2 (m + 1) (2m + 1) = 1 n k + 1-74-
(3) (k + 1) 2 k 2 = 2k + 1 = n 2 n (2) k, k + 1 n, k, k + 1 n 77 p, q, r, α, β x, y 1 { x + py = α qx + ry = β r pq \= 0 (1) x, y p, q, r, α, β (2) p, q, r r pq = 1 α, β x, y (3) r pq = 1 (α, β) = (1, 2) x 2 (p, q, r) 2006 (1) { x + py = α 1 qx + ry = β 2 1 x = α py 2 q (α py) + ry = β (r pq) y = β qα y = β qα r pq x = α p β qα r pq αr pβ = r pq (2) α = 0, β = 1 y = 1 r pq -75-
y r pq = 1 α, β x = ± (αr pβ), y = ± (β qα) (3) r pq = ±1 α = 1, β = 2 x = ± (r 2p), y = ± (2 q) ) x = 2 r 2p = ±2 r pq = 1 r 2p = 2 2p + 2 pq = 1 p (2 q) = 1 (p, 2 q) = ( 1, 1), (1, 1) (p, q) = ( 1, 1), (1, 3) (p, q, r) = ( 1, 1, 0), (1, 3, 4) r pq = 1 r 2p = 2 2p 2 pq = 1 p (2 q) = 1 (p, 2 q) = (1, 1), ( 1, 1) (p, q) = (1, 1), ( 1, 3) (p, q, r) = (1, 1, 0), ( 1, 3, 4) 2 p < q < r p, q, r 1 p + 1 q + 1 r 1 78 2010 2 p < q < r 1 p + 1 q + 1 r 1-76-
1 p > 1 q > 1 r 3 p > 1 p + 1 q + 1 r 1 3 > p p = 2 2 q > 1 q + 1 r 1 2 4 > q (> 2) q = 3 1 r 1 6 6 r r = 4, 5, 6 79 ABC C BC 3 p (1) AB,CA p (2) tan A tan B (1) AB=x,CA=y x 2 = y 2 + p 2 (x y) (x + y) = p 2 (2) p x, y x + y = p 2, x y = 1 x = p2 + 1, y = p2 1 2 2 tan A = p y = 2p p 2 1 = 2p (p 1) (p + 1) B x p A y C -77-
tan A tan B = y p = p2 1 (p 1) (p + 1) = 2p 2p p tan B 80 3 p p + 1 (1) 3 n = k 3 + 1 (k, n) (2) 3 n = k 2 40 (k, n) 2010 (1) 3 n = k 3 + 1 3 n = (k + 1) ( k 2 k + 1 ) k + 1 = 3 p k 2 k + 1 = 3 q p, q k 1 p 1 k 2 q 1 k (3 p 2) (3 p 1) + 1 = 3 q 3 2p 3 3 p + 3 = 3 q 3 2p 1 3 p + 1 = 3 q 1 3 1 q = 1 k 2 k 2 = 0 (k 2) (k + 1) = 0 k = 2 3 n = k 3 + 1 = 9 n = 2 (2) 3 n = k 2 40-78-
k k = 6m ± 1 3 n = (6m ± 1) 2 40 = 36m 2 ± 12m 39 3 n 1 = 12m 2 ± 4m 13 = 4 ( 3m 2 ± m 3 ) 1 3 n 1 = (4 1) n 1 = 4M + ( 1) n 1 n 1 n n = 2N ( 3 N ) 2 = k 2 40 ( k 3 N ) ( k + 3 N ) = 40 k 3 N, k + 3 N ( k 3 N, k + 3 N) = (2, 20), (4, 10) ( k, 3 N ) = (11, 9), (7, 3) (k, n) = (11, 4), (7, 2) m, n 3 ( n m n 2 + 1 ) l = 2 81 l, m, n 2010 ( n m n ) ( 1 2 + 1 l = 2 1 2 1 ) n = 2 m l m 3 1 6 1 2 1 m < 1 2 0 < 2 l = 1 ( 1 2 1 m n 6 ) n 1 1 6 n -79-
( i ) n = 3 ( 3 m 3 ) ( 3 2 + 1 l = 2 m 1 ) l = 2 2 6l ml = 4m ml 6l + 4m = 0 (m 6) (l + 4) = 24 (l + 4, m 6) = (8, 3), (12, 2), (24, 1) (l, m) (4, 3), (8, 4), (20, 5) ( ii ) n = 4 ( 4 m 4 ) ( ) 4 2 + 1 l = 2 m 1 l = 2 4l ml = 2m ml 4l + 2m = 0 (m 4) (l + 2) = 8 (l + 2, m 4) = (8, 1) (l, m) (6, 3) (iii) n = 5 ( 5 m 5 ) ( 5 2 + 1 l = 2 m 3 ) l = 2 2 10l 3ml = 4m 3ml 10l + 4m = 0 9ml 30l + 12m = 0 (3m 10) (3l + 4) = 40 (3l + 4, 3m 10) = (40, 1) (l, m) (12, 3) (iv) n = 6 ( 6 m 6 ) ( ) 6 2 + 1 l = 2 m 2 l = 2 3l ml = m ml 3l + m = 0 (m 3) (l + 1) = 3 l + 1 4-80-
82 a, b (1) ab 3 a b 3 (2) a + b ab 3 a b 3 (3) a + b a 2 + b 2 3 a b 3 2010 (1) a, b ab ab 3 a b 3 (2) ab 3 a b 3 a + b a b 3 (3) a 2 + b 2 = (a + b) 2 2ab 2ab ab a b 3-81-
83 n n = p 2 q p, q p \= q n 6 ( + p + p 2) ( ) + q n 2n 2n ( 2p 2 q = + p + p 2) ( ) + q + p + p 2 p + q 2p 2 + q = 2p2 k k ( q = + p + p 2) k q k = p 2 p = 0 p = n = 2010 1 1 1 2 2 28 n n = p 2 q p, q p \= q n 6 1 + p + p 2 + q + qp + qp 2 = ( 1 + p + p 2) (1 + q) n 2n 2n 2p 2 q = ( 1 + p + p 2) (1 + q) 1 + p + p 2 = 1 + p(p + 1) p 1 + q 2p 2 1 + q = 2p2 k k -82-
q = ( 1 + p + p 2) k q k = 1 p 2 p 2 = 0 p = 2 n=28 84 a, b, c, d c = 4a + 7b, d = 3a + 4b (1) c + 3d 5 2a + b 5 (2) a b c d p p = 5 2 1 2009 (1) { c = 4a + 7b d = 3a + 4b c + 3d = 13a + 19b = 15a + 20b (2a + b) c + 3d 5 2a + b 5 (2) 5a = 4c + 7d 5b = 3c 4d c d p p 5a, 5b a b 5a, 5b 5 p 5 p = 5-83-
x, y (1) 2 x + 1 y = 1 4 85 (x, y) (2) p 3 2 x + 1 y = 1 p (x, y) (x, y) 2x + 3y 2009 (1) 2 x + 1 y = 1 4 8y + 4x = xy (x 8) (y 4) = 32 (2) 2 x + 1 y = 1 p 2py + px = xy x 8 7, y 4 3 (x 8, y 4) = (1, 32), (2, 16), (4, 8), (8, 4), (16, 2), (32, 1) (x, y) = (9, 36), (10, 20), (12, 12), (16, 8), (24, 6), (40, 5) (x 2p) (y p) = 2p 2 (x 2p, y p) = ( 1, 2p 2), ( 2, p 2), (p, 2p), (2p, p), ( p 2, 2 ), ( 2p 2, 1 ) (x, y) = ( 1 + 2p, p + 2p 2), ( 2 + 2p, p + p 2), (3p, 3p), (4p, 2p), ( p 2 + 2p, p + 2 ), ( 2p 2 + 2p, p + 1 ) 2x + 3y A = 6p 2 + 7p + 2 B = 3p 2 + 7p + 4 C = 15p D = 14p E = 2p 2 + 7p + 6 F = 4p 2 + 7p + 3 p 3 A > F > B > E C > D -84-
E D = 2p 2 + 7p + 6 14p = 2p 2 7p + 6 2 3 2 7 3 + 6 > 0 (x, y) = (4p, 2p) 86 p n (p n )! p 2009 [ p n p ] + [ p n p 2 ] + + [ ] p n p n = p n 1 + p n 2 + + 1 = pn 1 p 1 n! p [ ] [ ] [ ] n n n + p p 2 + + p n + 87 a b 1 a n a n, b n (a + b 2) n = a n + b n 2 (1), (2) 2 (1) a 2 a 2 b 2 (2) n a n a n b n 2009 (1) ( a + b 2) n = an + b n 2 ( a + b ) n+1 ( 2 = a + b ) ( 2 a + b ) n 2 a n+1 + b n+1 2 = (a + b ) ) 2 (a n + b n 2 = a a n + 2b b n + (b a n + a b n ) 2-85-
a, b, a n, b n, a n+1, b n+1 { an+1 = a a n + 2b b n b n+1 = b a n + a b n a 2 = a 2 + 2b 2 b 2 = 2ab a a 2 a 2, b 2 g g b 2 = 2ab a, b a 2 = a 2 + 2b 2 g a 2, b 2 (2) a k a k+1 = a a k + 2b b k a k+1 a 1 = a a n ( a + b ) 2m ( 2 = a + b ) m ( 2 a + b ) m 2 a 2m + b 2m 2 = (a m + b m 2 ) 2 = am 2 + 2b m 2 + 2a m b m 2 { a2m = a m 2 + 2b m 2 b 2m = 2a m b m (1) a m, a m, b m a 2m, a 2m, b 2m a 2, a 2, b 2 a 2 n, a 2 n, b 2 n N a N, b N g a N+1, b N+1 g n N a n, b n N 2 m N < 2 m+1 m -86-
(1) x, y, z, a 175x = 1323y = 5832z = a 2 88 (2) a m 175, m 2 1323, m 3 m 5832 2009 (1) 175x = 1323y = 5832z = a 2 175 = 5 2 7, 1323 = 3 3 7 2, 5832 = 2 3 3 6 x = 7X 2, y = 3Y 2, z = 2Z 2 a = 5 7X = 3 2 7Y = 2 2 3 3 Z a 2 2, 3 3, 5, 7 a a = 2 2 3 3 5 7 = 3780 (2) m 175 = m 5 2 7, m 2 1323 = m2 3 3 7 2, m 3 5832 = m3 2 3 3 6 m 5 2 7 3 2 7 2 3 2 2 3 2 5 2 7 = 3150-87-
(1) x, y 1 < x < y ( 1 + 1 )( 1 + 1 ) = 5 x y 3 x, y (2) x, y, z 1 < x < y < z ( 1 + 1 )( 1 + 1 )( 1 + 1 ) = 12 x y z 5 x, y, z 89 2011 (1) ( 1 + 1 ) ( 1 + 1 ) = 5 x y 3 1 xy + 1 x + 1 y = 2 3 3 + 3y + 3x = 2xy ( x 3 ) ( y 3 ) = 9 2 2 4 + 3 2 (2x 3) (2y 3) = 15 1 < 2x 3 < 2y 3 (2x 3, 2y 3) = (1, 15), (3, 5) (x, y) = (2, 9), (3, 4) ( 5 3 = 1 + 1 ) ( 1 + 1 ) ( < 1 + 1 ) 2 x y x 5 3 < 1 + 1 15 + 3 x x < 2 x = 2, 3 x = 2 5 3 = ( 1 + 1 2 ) ( 1 + 1 ) y = 9 y -88-
x = 3 5 3 = ( 1 + 1 3 ) ( 1 + 1 ) y = 4 y (2) 12 5 = 12 3 ( 1 + 1 ) ( 1 + 1 ) ( 1 + 1 ) ( < 1 + 1 ) 3 x y z x 5 < 1 + 1 x x < 3 5 3 12 3 5 6 3 = 216, 7 3 = 343, 8 3 = 512, 9 3 = 729 3 3 5 3 12 3 5 = 5 ( 3 144 + 3 60 + 3 25 ) 12 5 3 720 + 3 300 + 5 = < 9 + 7 + 5 < 3 7 7 x = 2 ( 1 + 1 ) ( 1 + 1 ) ( 1 + 1 ) = 12 2 y z 5 ( 1 + 1 ) ( 1 + 1 ) = 8 y z 5 ( 8 5 < 1 + 1 ) 2 8 y 5 < 1 + 1 y y < 2 10 + 5 3 2 < y y = 3 ( 1 + 1 ) ( 1 + 1 ) = 8 3 z 5 z = 5 90 a, b a b > 0 x y 2 x 2 + ax + b = 0, y 2 + by + a = 0 (1) a = b a (2) a > b (a, b) 2011-89-
(1) x 2 + ax + a = 0 = a = α, β (α β) α + β = a, αβ = a a α + β = αβ (α + 1) (β + 1) = 1 (α + 1, β + 1) = (1, 1), ( 1, 1) (α, β) = (0, 0), ( 2, 2) a > 0 a = 4 (2) x 2 + ax + b = 0 α, β (α β) α + β = a, αβ = b a, b a > b > 0 α, β α β 1 a, b α β > αβ (α + 1) (β + 1) < 1 α β 2 α + 1 β + 1 1 (α + 1) (β + 1) 1 β = 1 1 a + b = 0 y 2 + by + a = 0 γ, δ γ + δ = b, γδ = a 1 a + b = 0 1 γδ γ δ = 0 (γ + 1) (δ + 1) = 2 {γ + 1, δ + 1} = { 1, 2} {γ, δ} = { 2, 3} a = 6, b = 5-90-
91 2 (1) 10 3 10 a, 1 b a + b b 0 4 (2) 10 5 1000 100 10 1 4 10000 2004 (1) 100M + 10p + q (100M + 10p + q) 2 = 100M + 20pq + q 2 q 2 b q 2 2pq a 2pq a q 2 a q 2 0 2 = 00, 1 2 = 01, 2 2 = 04, 3 2 = 09, 4 2 = 16 5 2 = 25, 6 2 = 36, 7 2 = 49, 8 2 = 64, 9 2 = 81 q q 2 a + b a + b q q 2 q = 0, 4, 8 b = 0, 4 (2) a + a = a = 0, 4 a = 4 (100M + 4) 2, (100M + 8) 2 (100M + 4) 2 = 10000M 2 + 800M + 16 (100M + 8) 2 = 10000M 2 + 1600M + 64 a = 0 (100M) 2 = 10000M 2 10000-91-
92 n, a, b 0 a, b ( ) a 2 + b 2 = 2 n (1) n 2 a, b ( ) a, b 0 (2) 0 n ( ) 0 (a, b) 2004 (1) a.b a 2 +b 2 n 2 a 2 +b 2 = 2 n a.b a 2 + b 2 = (2m + 1) 2 + (2n + 1) 2 = 4 ( m 2 + m + n 2 + n ) + 2 a.b (2) ( i ) n = 0 a 2 + b 2 = 1 (a, b) (a, b) = (1, 0), (0, 1) ( ii ) n = 1 a 2 + b 2 = 2 (a, b) (a, b) = (1, 1) (iii) n 2 a = 2m, b = 2n m 2 + n 2 = 2 n 2 n = 2p(p 1) a = 2 p a, b = 2 p b a 2 + b 2 = (2 p a ) 2 + (2 p b ) 2 = 2 2p (a ) 2 + (b ) 2 = 1 (a, b ) = (0, 1), (1, 0) (a, b) = ( ) ( 0, 2 n n ) 2, 2 2, 0 n = 2p + 1(p 0) a = 2 p a, b = 2 p b a 2 + b 2 = (2 p a ) 2 + (2 p b ) 2 = 2 2p+1 (a ) 2 + (b ) 2 = 2 (a, b ) = (1, 1) ) (a, b) = (2 n 1 n 1 2, 2 2-92-
93 (1) n n 3 + 1 3 (2) n n n + 1 3 2003 (1) n 3 n = (n 1)n(n + 1) n 3 + 1 = n 3 n + n + 1 n + 1 (2) f(n) = n n + 1 ( i ) n = 3m f (3m) = (3m) n + 1 ( ii ) n = 3m + 1 f (3m + 1) = (3m + 1) n + 1 = 3M + 1 + 1 = 3M + 2 (iii) n = 3m 1 f (3m 1) = (3m 1) n + 1 = 3M + ( 1) n + 1 n 3m 1 = 2l 1 m n = 3 (2k) 1 = 6k 1 n f(n), g(n) f (n) = n 7 ( 7 g(n) = 3f k=1 k n ) 94 (1) n f(n 7 ) = f(n) (2) n g(n) g(n) (2) 1995-93-
(1) (n 3) (n 2) (n 1) n (n + 1) (n + 2) (n + 3) = n ( n 2 1 ) ( n 2 4 ) ( n 2 9 ) = n ( n 6 14n 4 + 49n 2 36 ) = n 7 14n 5 + 49n 3 36n = n 7 n + 7 ( 2n 5 + 7n 3 5n ) (n 3) (n 2) (n 1) n (n + 1) (n + 2) (n + 3) n 7 n f(n 7 ) = f(n) (2) n r, q n = 6q + r k n k r = k 6q+r k r = k r ( k 6q 1 ) = k r ( k 6 1 ) { ( k 6 ) q 1 ( + k 6 ) q 2 ( + + k 6 ) } + 1 = k r 1 ( k 7 k ) { ( k 6 ) q 1 ( + k 6 ) q 2 ( + + k 6 ) } + 1 r = 1, 2, 3, 4, 5 f (k n ) = f (k r ) g (n) = 3f (1 + 2 n + 3 n + 4 n + 5 n + 6 n + 7 n ) = 3f (1 + 2 n + 3 n + 4 n + 5 n + 6 n ) = 3f (1 + 2 r + 3 r + 4 r + 5 r + 6 r ) 6 r + 1 = (7 1) r + 1 = 7M + ( 1) r + 1 5 r + 2 r = (7 2) r + 2 r = 7M + ( 2) r + 2 r 4 r + 3 r = (7 3) r + 3 r = 7M + ( 3) r + 3 r r g (n) = 3f (1 + 2 r + 3 r + 4 r + 5 r + 6 r ) = 0 r = 2 1 + 2 r + 3 r + 4 r + 5 r + 6 r = 7M + 2 (1 + 4 + 9) = 7N g (n) = 3f (1 + 2 r + 3 r + 4 r + 5 r + 6 r ) = 0 r = 4 1 + 2 r + 3 r + 4 r + 5 r + 6 r = 7M + 2 ( 1 + 4 2 + 9 2) = 7N -94-
g (n) = 3f (1 + 2 r + 3 r + 4 r + 5 r + 6 r ) = 0 r = 1, 2, 3, 4, 5 g(n) = 0 n = 6q 1 6q 1 (mod7) 2 6q = 8 2q 1 2q 1 (mod7) 3 6q = 27 2q ( 1) 2q 1 (mod7) 4 6q = 8 3q 1 3q 1 (mod7) 5 6q = 125 2q ( 1) 2q 1 (mod7) 6 6q = 36 3q 1 3q 1 (mod7) 7 6q 0 (mod7) g (n) = 3 6 = 18 95 (1) (2) 2002 (1) a, b, c a 2 + b 2 = c 2 r a r + b r = c r = a + b c 2 a, b, c r a, b, { (2k) 2 = 4k 2 (2k + 1) 2 = 4 ( k 2 + k ) + 1 a 2 + b 2 c 2 a, b, c r -95-
(2) 1 2 ab = a + b + c r 2 ab r = a + b + c abc rc = a + b + c 96 n 2 n 100 2 n 1 99 (1) n (2) 2 n (3) 2 n log 10 2 = 0.3010 2005 (1) 10 99 2 n < 10 100, 10 98 2 n 1 < 10 99 99 n log 10 2 < 100, 98 (n 1) log 10 2 < 99 99 log 10 2 n < 100 log 10 2, 98 log 10 2 + 1 n < 99 log 10 2 + 1 99 log 10 2 = 328.903 99 log 10 2 + 1 = 329.903 98 log 10 2 + 1 = 326.581 100 log 10 2 + 1 = 333.225 328.903 n < 329.903 n = 329-96-
(2) 2 5 2 (mod 10) 2 n+4 2 n (mod 10) 2 329 = 2 82 4+1 2 (mod 10) 2 239 (3) 2329 2 10 2 329 2 = 2 ( 2 82 4 1 ) 10 10 = 2 ( 16 82 1 ) 10 = 2 15 ( 16 81 + 16 80 + 16 80 + + 16 + 1 ) 10 = 3 ( 16 81 + 16 80 + 16 80 + + 16 + 1 ) 3 ( 6 81 + 6 80 + 6 80 + + 6 + 1 ) (mod 10) 3 (6 + 6 + 6 + + 6 + 1) (mod 10) = 3 (6 81 + 1) 3 (6 1 + 1) (mod 10) 1 (mod 10) 2 329 = 10936253623915059621862511135588 1068267658471544660621821288530320497 6499599687961611756588511526912 97 n f(n) = 5 3n + 5 2n + 5 n + 1 (1) n f(n) 13 (2) n f(n) 13 (1) n 5 2 = 25 1 (mod13) 5 3 5 (mod13) -97-
f (n) ( 5) n + ( 1) n + 5 n + 1 (mod13) f (4m + 1) ( 5) 4m+1 + ( 1) 4m+1 + 5 4m+1 + 1 (mod13) 0 (mod13) f (4m + 3) ( 5) 4m+3 + ( 1) 4m+3 + 5 4m+3 + 1 (mod13) 0 (mod13) f (4m + 2) 25 2m+1 + 1 + 25 2m+1 + 1 (mod13) ( 1) 2m+1 + 1 + ( 1) 2m+1 + 1 (mod13) 0 (mod13) n f(n) 13 (2) f (4m) ( 5) 4m + ( 1) 4m + 5 4m + 1 (mod13) 25 2m + 1 + 25 2m + 1 (mod13) ( 1) 2m + 1 + ( 1) 2m + 1 (mod13) 4 (mod13) n f(n) 13 98 p, q 1 < p < q L L = { (m, n) m, n 0 m < q 1, 0 n < p 1 } L A(m, n) N(A) = mp + nq (1) L A, B N(A) = N(B) A = B (2) L A(m n) L A (q 2 m, p 2 n) A \= A (3) N(A) pq (p + q) N(A ) pq (p + q) (4) N(A) pq (p + q) L A 2002 (1) A(m, n),b(r, s) N(A) = N(B), mp + nq = rp + sq (m r) p = (s n) q -98-
p, q m r q (q 1) m r q 1 m r = 0 n s = 0 A = B (2) A = A m = q 2 m, n = p 2 n m = q 2 1, n = p 2 1 m, n p, q p, q A \= A (3) N (A) pq (p + q) mp + nq pq (p + q) M = q 2 m, N = p 2 n m = q 2 M, n = p 2 N (q 2 M) p + (p 2 N) q pq (p + q) Mp + Nq pq (p + q) N ( A ) pq (p + q) (4) L A A N (A) pq (p + q), N (A) pq (p + q) L (p 1)(q 1) N (A) pq (p + q) A (p 1)(q 1) 2 99 (1) 5 n 6n+1 6n 1 (2) N 6N 1 6n 1 n (3) 6n 1 n 2009-99-
(1) 2,3 6n + 1, 6n 1 (2) 6N 1 5 2 3 6n + 1, 6n 1 6n i + 1 6N 1 = (6n 1 + 1) a (6n k + 1) b 6 6n 1 (3) 6n 1 6n 1 1, 6n 2 1,, 6n M 1 1 P P = 6Q + 1, 6Q 1 P = 6Q + 1 P 2 = 6Q 1 1 P 2 P = 6Q 1 P 6 = 6(Q + 1) 1 1 P 6 = 2 3 6n 1 n -100-