1 11 111 111-1 p, q, r A, B, C (1 p<q<r) 3, A, B, C 20, 10, 9 Br, q? p, q, r s = p + q + r A B C (1) s (2) s (3) r s 20 10 9 3s A + B + C =20+10+9=3s s = 13 3r 20, r 7 B = 10 Br B =10 r +2p 7+2p p =1 2r + p, r +2q, r + p + q B =10

116 B = r +2p r =8 (p, q, r) =(1, 4, 8) A B C (1) 1 (2) 1 (3) 8 20 10 9 A =20 8+8+4, 8B A, 8 C A B C (1) 8 1 4 (2) 8 1 4 (3) 4 8 1, q =4 : C,, 111-2 ab, a 2a 3a (b 1)a (a 1)(b 1) + + + + = b b b b 2 1 0 <x<b,0<y<a (a 1)(b 1), y =(a/b)x ( ) a, b

11 117, 2 S a 2a (b 2)a (b 1)a S = + + + + b b b b (b 1)a (b 2)a 2a a S = + + + + b b b b b 1 ia (b i)a b 1 ia 2S = + = b b b i=1 b 1 = (a 1) = (a 1)(b 1) i=1 i=1 (b i)a + 1 b ( ) ( ) x + y = n ( ) x, y [x + y] =[x]+[y]+1, 111-3 a 1,a 2,a 3, m 1, b m =min{ n : a n m }, b ma n m n a 19 =85, a 1 + a 2 + + a 19 + b 1 + b 2 + + b 85 a p = q, p q i a i, a i a 1 a 2 a 3 a 4 a 5 a 6 a 7 j j a i, b j 1, p q a i + (b j 1) = ( )+( )=pq i=1 j=1, (p +1)q

118 a 1,a 2,,a 19,, 85 i=1 b i = b 1 + b 2 + + b 85 b 1,,b a1 =1, b a1 +1,,b a2 =2, b a18+1,,b a19 =19 =1 a 1 +2 (a 2 a 1 )+ +19 (a 19 a 18 ) = a 1 a 2 a 18 +19a 19 = 19 j=1 a j +20a 19, 19 a j + 85 j=1 i=1 b i =20a 19 = 1700 a p = q, (p +1)q 112 112-1 ABC O A BC D, D BC AO E ADE, B = C AD M

11 119 M BC, OMBC, OM//DE EAD OAM, OAMOA = OM ADE AE = DE 112-2 ABC = CDE =90 ABCDE AE M, AB CD = BC DE MB = MD 1 ACCE P, Q, MPA = EQM, AP B =2 ACB, EQD =2 ECD, MP = EC/2 =DQ, MQ = AC/2 =BP AB BC = DE CD ACB = ECD AP B = DQE MPB = DQM MPB DQM MB = MD 2 AB = BP P AB, ED = DQ Q ED ACP =2 ACB, ECQ =2 ECD, AC = CP, EC = CQ, AB = BP, AM = ME, EP =2MB, AQ =2MD

120, AB BC = DE CD ACB = ECD ACP = ECQ ECP = ACQ ECP ACQ EP = AQ MB = MD 3 DM DM = MX X AM = ME, AMX = EMD AMX EMD BA BC, AX DE, DE CD AX CD B 90 BAX = BCD AB DE = BC CD BA AX = BC BAX BCD CD ABX = CBD XBD =90 MB = MD = MX 113 113-1, 38 55, 50 55?

11 121 a, b, p, q, x, y k = y/x a + kb = p, ka + b = q x 2 = a 2 + b 2 (k 2 1) 2 x 2 =(k 2 1) 2 a 2 +(k 2 1)b 2 =(kq p) 2 +(kp q) 2 p q =38 55, 50 55 (55k 38) 2 +(38k 55) 2 =(55k 50) 2 +(50k 55) 2 2k 2 5k +2=0, (2k 1)(k 2) = 0, y =2x, a+2b =50,2a+b =55 (a, b) =(20, 15), x 2 = a 2 +b 2 (x, y) =(25, 50), 113-2, 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 k, L, R L 1:k R

122, 5 (1) A + L = k(a 1 + R) (2) B + L = k(b 1 + R) (4) A 2 + L = k(a + R) (5) B 2 + L = k(b + R) (3) C + L = k(c 1 + R) 5, 6(A, B, C, k, L, R), LR L kr 5 (1) (2)(4) (5) (A B) =k(a 1 B 1 ), (A 2 B 2 )=k(a B), k 2 = A 2 B 2 A B k (4) (1), (5) (2) A, B,(3) (1)+(2) 2 A = ka 1 + A 2, B = kb 1 + B 2 k +1 k +1 C = A + B 2 + kc 1 k A 1 + B 1 2 k, A, B C = kc 1 + k(a 1 + B 1 )+(A 2 + B 2 ) k(a 1 + B 1 )(k +1) 2(k +1) = kc 1 + (A 2 + B 2 ) k 2 (A 1 + B 1 ) 2(k +1) A2 B 2 = C 1 + (A 2 + B 2 ) A2 B2 A 1 B 1 (A 1 + B 1 ) A 1 B 1 A 2( 2 B 2 A 1 B 1 +1) A2 B 2 = C 1 + (A 2 + B 2 )(A 1 B 1 ) (A 2 B 2 )(A 1 + B 1 ) A 1 B 1 2( A 2 B 2 A1 B 1 +(A 1 B 1 )) A2 B 2 A 1 B 2 A 2 B = C 1 + 1 A 1 B 1 A2 B 2 A1 B 1 +(A 1 B 1 ), 114

12 123 114-1 3 3 n 3 n,, 3 n 2, 1 2 2 13, 1 1 n 1 (0 ), n 2 n/2,, 114-2 A =(0, 0), B =(0, 1), C =(1, 0) A, B, C, D =(1, 1)? (, ),, D,, 12 121

124 121-1,,? Aa, Bb, Aa Bb () ab (AB) ab () AB ab (A) B!!, Aa A A BBb A b B B b Aa Bb () Bb (Aa) Bb () Aa Bb (A) a ABb () a b (AB) a b () Aa B b (B) Aa Bb () Aa (Bb) Aa,, 2

12 125 121-2 n,3n +1 n +1 3n +1 = s 2 s3, s =3t ± 1 n = s2 1 3 =3t 2 ± 2t n +1=3t 2 ± 2t +1=t 2 + t 2 +(t ± 1) 2 121-3 f(n) n : 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, f(s + t) f(s t) =st, s, ts >t n f(n), s + t s t 1 {a n } f(n) =a 1 + + a n f(s + t) f(s t) a s t+1 a s+t(2t), a n + a n+1 = n f(s + t) f(s t) =(a s t+1 + a s t+2 )+ +(a s+t 1 + a s+t ) =(s t +1)+(s t +3)+ +(s + t 1) = t[(s t +1)+(s + t 1)] 2 ( ) ( + ) 2 = st 122

126 122-1 X (1) 1 X (2) u, v X = uv u + v X X Y =1/X = { 1 x : x X}, x X 1 x Y 1 uv u+v = u + v uv = 1 u + 1 v, (1 )1 Y (2 ) 1 u, 1 v Y = 1 u + 1 v Y 1 u, 1 v p, q (2 ) (2 ) p, q Y = p + q Y Y Y = N, X X =1/N = { 1 n : n N} 122-2 x 3 +2y 3 +4z 3 +8xyz =0 f(x, y, z) =x 3 +2y 3 +4z 3 +8xyz f(x) =0 x 3 x x =2x 1 f(x, y, z) =8x 3 1 +2y 3 +4z 3 +16x 1 yz =2(y 3 +2z 3 +4x 3 1 +8yzx 1 )=2f(y, z, x 1 ), 0=f(x, y, z) =f(y, z, x 1 )=f(z,x 1,y 1 )=f(x 1,y 1,z 1 ), (x, y, z) =2(x 1,y 1,z 1 ) n (x, y, z) =2 n (x n,y n,z n ),0 2 (0, 0, 0)

13 127 13 131 131-1 a, b, c, A, B, C ac 2bB + ca =0 ac b 2 > 0, AC B 2 0 AC B 2 > 0 0 (ac ca) 2 =(ac + ca) 2 4acAC =4b 2 B 2 4acAC < 0, AC B 2 0 131-2 ( ), p, q, 3 r p, 3 3 q, r 3 q 3 p 3 r 3 q = n (m, n ) m (m + n) 3 q = m 3 p + n 3 r (m + n) 3 q = m 3 p + n 3 r +3mn 3 pr(m 3 p + n 3 r ) = m 3 p + n 3 r +3mn(m + n) 3 pqr 3 pqr 132

128 132-1 S2002 N0 N 2 2002 S, (a) (b) (c) N S S = M, M, 0 N 2 M N S (1) M =0 ; S 2 M =1, N =0, 1, (2) S = M, 0 N 2 M S = M +1,0 N 2 M+1 S a, S = S \{a} ( S = M) (i) 0 N < 2 M ; S N = N / S a S S a (ii) 2 M N 2 M+1 ; S N = N 2 M / S a S S a (2 M ) N (c) (a) (b) (i) S (a ),() () S S (a ), () S () () S,

13 129 ()a () () a, (i) (a), (b) (ii) (a), (b) S (1), (2) S = M 0 N 132-2 S n, S 1 S n/7, 3 S T n 7 T S <n, S = n xy-, y (P ) 3, 1, 1 3 O 1 ( ), O 2 ( ) 6 ( ), S S n 7 S <n, S ( n 7 1) n S 7

130 ( ) 30 6 7 2 ( ) 1, 6 132-3 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 1 m =1, m =2 Cauchy-Schwarz m =4 m =2 2 2 n n n n n n a 4 j b 4 j c 4 j d 4 j a 2 jb 2 j c 2 jd 2 j j=1 j=1 j=1 j=1 j=1 j=1 2 2 n (a j b j )(c j d j ) j=1, m =8 m =4 m =2, m =2 k+1 m =2 k m =2, Cauchy 2 k m 2, m2 k a i,j m m m n n a i,j g j ( ) i=1 j=1, g j a 1,j,,a m,j, g j = m m i=1 j=1 a i,j

13 131 m =3 m =4 4 4 n n n n n n g j (a j b j c j g j ) 1/4 = a j b j c j j=1 j=1 j=1 j=1 j=1 g j j=1 m m2 k t t = m + r a 1,1,,a 1,n ; ; a m,1,,a m,n a x,j = g j (m<x t) a 1,1,,a 1,n ; ; a t,1,,a t,n t ( ) t n n a i,j t t t i=1 j=1 j=1, r m n n n m t a i,j i=1 j=1 g j j=1 m n a i,j i=1 j=1 j=1 n g j j=1 i=1 m i=1 a i,j t n a i,j [gj r] = g j j=1 m t 133 (recurrence) 133-1 (2002KMO) 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

132 a n x ny n a n = a n 1, x x n 1 y n 1 a n =2a n 2, a n = a n 1 +2a n 2, a 0 =1, a 1 =3, a n 2a n 1 = (a n 1 2a n 2 ) =( 1) n 1 (a 1 2a 0 )=( 1) n 1, a n =2a n 1 +( 1) n 1 a n + 1 3 ( 1)n =2 a n 1 + 13 ( 1)n 1, a n = 4 2n ( 1) n 3 a n =2 n a 0 + 1 3 ( 1)0 1 3 ( 1)n = 4 2n ( 1) n 3 133-2 3 n,? 1 n a n a n+1n n n 1 2 n (, 1) (1, 2),, (n 1, n), (n, ) (n +1)

13 133 a n+1 = a n + n +1, a 1 =2 a n = 1 2 n2 + 1 2 n +1 2 c n c n+1c n n n n c n+1 = c n + a n, c 1 =2 c n = 1 6 n3 + 5 6 n +1 133-3 A, B A, B Am, Bn ( )?, 134 134-1 S{1, 2,,9}, S {1, 2, 3, 5} {1, 2, 3, 4, 5} 1+4=2+3 S? 1 1 9 3 17 15 S n, n 15 n 6 n 2 (1) n =6 : 6 2 =15 3 17 3

134 {1, 2} S, 17 {8, 9} S, 1+9=2+8 (2) n =5 :, {1, 2, 5, 7, 9} 5 2 {1, 2, 3, 5, 8} n 6 S6 a 1 >a 2 > >a 6 i, j, k, l a i a j = a k a l a i + a l = a j + a k, a 1 a 2, a 3 a 4, a 5 a 6, a 2 a 3 a 4 a 5, (a 1 a 2 )+(a 3 a 4 )+(a 5 a 6 ) 1+2+3=6 (a 2 a 3 )+(a 4 a 5 ) 1+2=3 a 1 a 6 9, a 1 a 6 9 1=8 n =5 134-2 M = { (x, y) 1 x 12, 1 y 13, and x, y Z } (1) M 49, x, y (2) 48 (1) (1) (2) 134-3 n, 1/n n (n +1)/2

2 21 211 211-1 O O AB XY, AX BY P 211-2 {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 np n (x) n,2 k n S n ( ) k n

136 (1) P n+1 =2xP n (x) P n 1 (x) P 3 (x)p 1 (x) =(2xP 2 (x) P 1 (x))p 1 (x) =(P 2 (x)) 2 (P 1 (x)) 2 n =2 n (n 3) P n+1 (x)p n 1 (x) =P n (x) 2 P 1 (x) 2 = P n (x)(2xp n 1 (x) P n 2 (x)) P 1 (x) 2 =2xP n (x)p n 1 (x) (P n (x)p n 2 (x)+p 1 (x) 2 ) =2xP n (x)p n 1 (x) P n 1 (x) 2 P n+1 (x) =2xP n (x) P n 1 (x), n, n (1) (2) 2xP n (x)p n 1 (x) P n (x)(n +1), n,, (+),, (2) P 1 (x) =x 2 1, P 2 (x) =2x 3 2x (1) P n+1 (x) =2xP n (x) P n 1 (x), n +2 n 2xP n + + P n 1 + P n+1 (n +2),,,(2) (3) S 1 =2,S 2 =4,S n+1 =2S n + S n 1 (2) (4) k n =2L(n), L(n)n 2, k n = L(S n ), S n = 1 (1 + 2) n (1 2) n 2 n =2 1 n n 2 2 2 i i i=0 i=0 = n 2 i+1 2 i 0 i n i:odd n ( 1) i 2 2 i i

21 137 n i =2r +1 = n n 1 2r +1 2r +1 2r S n = n 2 r+1 =2n + n 1 2r 2n 2r +1 2r 2r +1 0 r n 1 2 1 r n 1 2, 1 r n 1 2 2 L(2n)+1 2 r 2n, 2 L(2n)+1 n 1 2r 2 L(2n)+1 1 r n 1 2 2n =2 L(2n) (2t +1)(t N), n 1 2r 1 r n 1 2 2r 2n 2r +1 2 L(2n)+1 n 1 2r 2n, 2r +1 n 1 2r 2r 1 r n 1 2 2r 2n 2r +1 2r 2n 2r +1 =2L(2n)+1 s, S n =2 L(2n) (2t +1)+2 L(2n)+1 s =2 L(2n) (2(t + s)+1) L(S n )=L(2n) =2L(n) 211-3 {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 1 1 + (ii) a n = a n a 1, a n 2 = (a n ) 2, a n2 k = a n 2a k a 1 = a 4 = = a n 2 = =+, a n2 k = a k a 2 =, a 5 = a 2 =+ (iii) a 3 =, (ii) a 6 = a 8 = a 9 =+ a 7 = a 10 = a 10 = a 5 a 2 a 5 =, ++ + ++

138, (iii) a 2 = a 5 =+ a 4 =+(iii) a 3 = a 6 =, a 6 = a 3 a 2 a 5 = [ ] k a 3k+1 =+,a 3k+2 = a = a 3 a r : n =3 r (3k +1) a n = a r : n =3 r (3k +2), (ii)(iii), a 2 k =0, 1 a 1 = a 4 =+,a 2 = a 5 = 0 k<2l (l 1) k =2l, 2l +1 a 6l+2 = a 2 a 3l+1 =, a 6l+4 = a 2 a 3l+2 =+ a 6l+1 =+, a 6l+5 = a 6l+1 = a 6l+2 = (iii) a 6l = a 6l+3 =+, a 2l = a 2l+1 = a 3 a 4l = a 4l+2 = a 3 (iii) a 4l+1 = a 3, a 12l+3 =+ a 12l+2 = a 12l+3 = a 12l+4 =+ (iii) a 6l+1 =+ a 6l+5 =+ a 6l+3 = a 6l+6 =, a 2l+1 = a 2l+2 = a 3, a 4l+2 = a 4l+4 = a 3, a 4l+3 = a 3, a 12l+9 =, a 12l+8 = a 12l+9 = a 12l+10 = (iii) a 6l+5 = 212 212-1 x, y, z, (x + y + z) 2 a(x 2 + y 2 + z 2 ), a

22 139 212-2 (m, n) : (1 + x n + x 2n + + x mn ) (1 + x + x 2 + + x m ) 22 221, 221-1 n a 1,,a n, 8 7 1 + + 1 20 a 1 a n 221-2 2000 n n k, xx n=1 k=1 221-3 n,3 n +1, 1

140, ( ) 222 222-1 A, B X, AX BX X? B B B, A B X 222-2 13,,? 222-3 10 37, 23 37

22 141 223, 223-1 n 4 n n n, m (, m<n) m m (m 2) m<n m, (m 2), 223-2 a, b > 0, a, A 1,A 2,b, a, G 1,G 2,b A 1 A 2 G 1 G 2, 224

142 224-1 6, 8, 10 224-2 n, n ( ), n 1 (n 1) X n X X A, B, C, D, A X X ( ) P ( ) X P, n P 2, AB AD

22 143 BD, BC 1, CD 1 2 BC1 P, CD1 Q C X d ( ) X BC d/2 P, AB d/2 Q [( )( ) ] X,,, (: ( 1 3, 2)), 224-3 1 n, 1 n n f(x) 1 n n =0 f(x) 1 f(x) = 1+1 2, n =1 f(x) x + a x x +2a n 2 a 1 <a 2 < <a n 1 n 1 g(x) =(x a 1 )(x a 2 ) (x a n 1 )

144 n 1 g(x) M a(x) =f(x)+mg(x), b(x) =f(x) Mg(x) a(x)b(x) f(x), 1n, a 1,,a n 1 g(x) ( ) n ( ), a 1,,a n 1 b 1 <a 1 <b 2 <a 2 < <b n 1 <a n 1 <b n b 1,,b n g(b 1 ),,g(b n ) [b 1,b n ] f(x) A, g(b i ) B, A< MB M a(b 1 ),,a(b n ) b(b 1 ),,b(b n ) a(x) x f(x)mg(x) b(x) x f(x) Mg(x), x ± Mg(x) f(x), a( ), a(b 1 ),, a(b n ) b(b 1 ),, b(b n ),b( ) a(x)b(x) n 225, 225-1 4n +3 1 4n +3, p p! 1 N = p! 1=p 1 p 2 p m, p i p 4n +1, p i p p! p!, 1 N, N4n +1 4n +1 p! 4, N4n +3, 4n +3

22 145 2 4n +3, k, k 4n +3 p 1,p 2,,p k P = p 2 1 p2 2 p2 k +2 (1) P 4 3 p 2 i 4 1 (2) P p 1,p 2,,p k, p ip 2 1p 2 2 p 2 k, p ip p i2, p i 2, P 4n +3,4n +1 4n +1 4n +1,4n +3 (1) 225-2 a a n nlim =0 i =1, 2,,n 1 n a n i + a n+i < 2a n n? M n M a n 1 + a n+1 2a n a n+1 a n a n a n 1, α = a M a M 1 a M+1 a M a M+2 a M+1, t, a M+t a M+t 1 + α a M+t 2 +2α a M + tα a M+t M + t a M+t 0 = lim lim t M + t t a M t M + t α M + t + a M+t M + t + 1 M +1 α = α! t

146 23 231, 231-1 6,,, 6? 6 0, 1,,5, a 1,a 2,,a 6 modulo 6 a i 1 1 a i S S =0+1+ +5 3 (mod 6) 6 i S =6 i 0 (mod 6), S modulo 6 231-2 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 n4 S 1 1 a 1,a 2,,a n,4 n S 4, S =0, S4 a i +, a i a i, 4

23 147 a i a i a i S4, S4 (+, ) (4, 0) (0, 4) 4 4 (3, 1) (1, 3) 2 2 (2, 2) (2, 2) S =0,4 S, a i + 4 S a i 1 S 1, n S = n 4 n 231-3 100a 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, i =20, 40, 60, 80, 100 a i = 1 i 5 a i =1,?,,, a n, a m a n, a m, a n n + a m a 2 m = n a2 m + a m a 2 + m a2 n + a n na m ma n m ma n na m n = a n n + a m m, S = a 1 1 + a 2 2 + + a 100 100 (a n,a m ) (a n,a m)

148 S = 1 1 5 20 + 1 40 + + 1 + 100 1 i 100 20i 1 i ( ) S = 100 i=1 a i i, a i Z ( ) 5,( ) 5 3,( ) 5 2, 232 232-1 x, n, [nx] > [x] 1 + [2x] 2 + [3x] [nx] + + 3 n [t]t (: [π] =3,[ 2]=1) 232-2 S 1, 0, 1 f : S Sf(x) = x 1 x f (1) (S) f (2) (S) f (n) (S) = φ (, f (n) fn ) f q = x2 1 = q2 p 2 p x pq q 2 p 2 pq f(x) x 1 (q =1 f p = 1 p2 p 1 ) f(s) f 2 (s) = φ

24 149 232-3 x i, s = x i > 0 x, y, z y<0,(x, y, z)(x + y, y, y + z) y<0? 5 f(x 1,x 2,x 3,x 4,x 5 )= (x i x i+2 ) 2, i=1 x6 = x 1 x 7 = x 2 y = x 4 < 0,(x 3,x 4,x 5 )=(x 3 + x 4, x 4,x 5 + x 4 ) f, f f = = 2 5 (x i x i+2) 2 i=1 5 (x i2 x 2 i ) 2 i=1 5 (x i x i+2 ) 2 i=1 5 (x ix i+2 x i x i+2 ) i=1 = 2(2x 2 4 +2x 3 x 4 +2x 5 x 4 ) = 2sx 4 < 0 2(x 1 x 4 2x 2 x 4 +(x 3 + x 4 + x 5 )x 4 2x 1 x 4 + x 2 x 4 ), f f, f f, 11, 10 24 241 241-1

150 1 n, a + b =2c a, b, c n 241-2 n n, 1 1 4 2,2 242, 242-1? n 1, n, n +1 (n 1) 4 + n 4 +(n +1) 4 =3n 4 +12n 2 +2 3k +2 3 01, 242-2 {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

25 151 242-3 n,6 n +1,n+2,n+3,n+4,n+5,n+6 A, B, A B 7 25 251 251-1 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) (ii) f(30) = f(2)f(3)f(5) f(2), f(3), f(5) () f f(n) n,2 4 =4 2 (iii) f(2) = 4 () f(3) f(8) <f(9) f(3) 2 >f(2) 3 =8 2, f(3) > 8 f(243) < f(256) f(3) 5 <f(2) 8 = 65536 < 10 5, f(3) < 10 9, f(3) = () f(5) f(24) <f(25) f(5) 2 >f(2) 3 f(3) = 24 2, f(5) > 24 f(125) <f(128) f(5) 3 <f(2) 7 = 16384 < 26 2, f(5) < 26, f(5) = 25 f(30) = f(2)f(3)f(5) = 900 f(n) n 2?

152 251-2 a 1 a 2,a 3,, 12345, 123457, 1234571, 12345718,,9,, (2 )5(5 ) 1, 3, 7,17mod 3 1 3mod 3 0,1 73 mod 3 1 3, 1 7,3 a 1 = p (> 10), a n+1 =10a n +3, mod p, a 1 0(modp), p10 a k 0(modp), a k, 252 252-1 3 3 3 1, 2, 3,,27,,, 3?

25 153 1, 2, 3 0 1 2 6 8 1 1 2 0 9+ 7 3 5 2 0 1 2 4 9 2 0 1 8 1 6 0 1 2 9+ 3 5 7 1 2 0 4 9 2 1 2 0 1 6 8 2 0 1 9+ 5 7 3 0 1 2 9 2 4 252-2 a, b2 k n 1,n 2,,n k (a) n 1 = a, n k = b, (b) n i n i+1n i + n i+1 (1 i<k) a b ab (i) a a (ii) a b b a (iii) a b, b c = a c (i)k =1, (ii) n 1,n 2,,n k (iii)a b b c a = n 1,,n k = b = n 1,,n k = c, m i = 1 (b) n in i+1 n i n i + n i+1 1 n i n i+1 = n i + n i+1 = 1 + 1 n i +n i+1 n i n i+1 n i n i+1 1, 1 n n m 1,m 2,,m k

154 (a ) m 1 = 1 a, m k = 1 b, (b ) m i + m i+1 1 n (1 i<k) (1) n n(n 1) 1 n + 1 n(n 1) = 1 n 1 (1 ) kn n(n k) 1 kn + 1 n(n k) = 1 n k (, n>k) 1 (2) n(n 1) n(n 1)(n 2) n(n 1) + 1 n(n 1)(n 2) = 1 n(n 2) (2 ) n(n 1) (n k) n(n 1) (n k 1) (, n>k+1) (3) n(n 1)(n 2) n(n 2) 1 n(n 1)(n 2) + 1 n(n 2) = 1 (n 1)(n 2) (4) n 2n n n(n 1) n(n 1)(n 2) n(n 2) 2n (5) 2n(n 1) (n +1)n(n 1)(n 2) (n +1)(n 2) + 2 = n(n 1), n (1) n(n 1) (4) 2n(n 1) (5) (n +1)n(n 1)(n 2) (2 ) (n +1)n(n 1) (2) (n +1)n (1) n +1 n n +1 a<b a a +1 a +2 b 26 261 261-1 R, ar 2 + br + c dr 2 + er + f 3 2 < R 3 2 a, b, c, d, e, f

26 155 1 R 3 2 0, R = 3 2 0, a 3 4+b 3 2+c =2d + e 3 4+f 3 2 a, b, c, d, e, f a = e, b = f, c =2d 2, R 3 2, er 2 + fr +2d dr 2 + er + f 3 2 < R 3 2 (e d 3 2)R 2 +(f e 3 2)R +(2d f 3 2) dr 2 < R 3 2 + er + f (R 3 2) (e d 3 2)R +(f d 3 4) dr 2 < R 3 2 + er + f, R 3 2 (e d 3 2)R +(f d 3 4) < dr 2 + er + f, 3 R>0, d, e, f, (e d 3 2)R +(f d 3 4) <er+ f<dr 2 + er + f, e>d 3 2, f > d 3 4, (d, e, f) =(1, 2, 2)

156 261-2 n m, m m,, I,, m, n 6 2m <n (a) n, I m, n (b) I n, m =3 n I = n (a) mn n m, I, ( ) m 1 2 2(m 1) I max = 1 n 2(m 1) = n(m 1) 2 (b) I n P i 2, Q i (i =1, 2,,n), P i = P j Q i = Q j ( ), P i Q i ( ) I n ( ) : P i = P j, Q i = Q j Q i = Q j P ip j, P j P i Q i, P j P ip j P j P i nn =2q I = n q A A q B, AB

26 157 2q 2q, q A, B I 262 262-1 a 2 + b 2 + c 2 = a 2 b 2 262-2 c/d (d <100) k c = k 73 (k =1, 2, 3,,99) d 100, [x]x 262-3 2n, n, n n,? 262-4 n 1,n 2,n 3, n k+1 >n nk (k 1) 1, 2, 3, 4,

158,? n k n k n k k, n k+1 >n nk k +1>n k, n k = k? n k+1 n nk, a 2,a 3,a 4, n 2,n 3, n k+1 >n nk, n 1 n 1 (n 1 ) n 1 n 2 1,n 3 1,n 4 1, n k = n k+1 1 n k+1 = n (k+1)+1 1 >n nk+1 1=n n k +1 1=n n k,, n 1 = n 2 1, n 2n 3,n 4,,, n k, 263 263-1 P, Q ABCD, PAQ < 60

26 159 263-2, ( ) 263-3 A, B, C, D, AB, AC, AD, BC, BD, CD 1

160

3 31 311 311-1 x 7 2x 5 +10x 2 1=01 311-2 a, b, c, d, a 2 + b 2 =2 c 2 + d 2 =2 ac = bd a 2 + c 2 =2 b 2 + d 2 =2 ab = cd a 2 c 2 = b 2 d 2 =(2 a 2 )(2 c 2 )

162 a 2 + c 2 =2 b 2 + d 2 =2 b 2 = c 2, a 2 = d 2 a 2 b 2 = c 2 d 2, ab = cd, ac = bd a, b, c, d ab = cd, (a 2 + b 2 2) 2 +(c 2 + d 2 2) 2 +2(ac bd) 2 =(a 2 + c 2 2) 2 +(b 2 + d 2 2) 2 +2(ab cd) 2, 0 0, 311-3 n 3 3 1/4 ( ), (3 ), WLOG, 312 122-1 1 x, 1 y, 1 z (xyz 1) x, y, z,?

31 163 xyz, yz, zx, xy 1 xyz = yz + zx + xy +1, 1 = 1 x + 1 y + 1 z + 1 xyz x y z x =1 x 2 1 x 1 y 1 z > 1 xyz, 1= 1 x + 1 y + 1 z + 1 xyz < 1 x + 1 x + 1 x + 1 x = 4 x x<4 x = 2 or 3 x =3 2 3 = 1 y + 1 z + 1 3yz < 1 y + 1 y + 1 3y = 7 3y 1 2y <7, y =3 3 = 1 z + 1 9z = 10 9z, z = 10 3 x =2 1 2 = 1 y + 1 z + 1 = yz =2z +2y +1 = (y 2)(z 2) = 5 2yz (y 2,z 2) = (1, 5), (x, y, z) =(2, 3, 7) xyz 1=41 312-2 n 3 t(n), n ( ) : t(3) = 1 : n n t(2n 1) t(2n) = +1 6 6 122-2 x 3 +2y 3 +4z 3 + 2003xyz =0 (x, y, z) =(0, 0, 0) mod7 n n 3 0, ±1 (mod7) (1) (x, y, z)(0, 0, 0) x + y + z (A) x, y, z 7 2y 3 +4z 3 0 (mod 7) x 3 +4z 3 0 (mod 7) x 3 +2y 3 0 (mod 7)

164 (1) mod 7 6 7 x 3 y 3 0, x, y, z 7 ( x 7, y 7, z 7 ), (x, y, z), x, y, z 7 (B) x 3,y 3,z 3 ±1(mod7) n 1, 2, 3 n 3 1 (2) z 3 1 x, y, z, z 3 1 2003 1(mod7) mod 7 x 3 +2y 3 +4+xyz 0 (mod 7) x 3 y 3 1 xyz 0, (x 3,y 3 ) ( 1, 1), ( 1, 1), (1, 1) xyz 1, 2, 3 n 3 1, x 3 y 3 z 3 1, x, y, z 1, 2, 3 xyz 1, 2, 3 1 x 3 y 3 z 3 (xyz) 3 1 WLOG,, 32 321 321-1 P P,, v, e, f e 3v 6, 2e, 3, 3f, 2e 3f

32 165 P v e + f =2 f =2 v + e 2e 3(2 v + e), e 3v 6 321-2 T,,,, T 1, 2, 3 1, 2, 3, T, 1, 2, 3 [ ] 1, 2, 3 T i 12-(12 ) a i, a = a i T 12- a, T 12- a T 3 12-, 12 12-, a 12-123- 12-, 112-122- 12-, 12-, a 123-,, 322

166 322-1 0 <m<3 y =(m +1)x m 2 322-2 1 1000 1000,1 1000,,,, 1000?,, [ ] n>2, n n, 1000 d>2 (0,d] d d + kd kd, (0,d], (d, 2d], (2d, 3d],, (1000 d, 1000] d 1000 d d 1000 12 1 1000,2 500,, 0 [ ]

32 167 n>2 n = 1 n k [1,n/2) nk nn k(= k ), k k (n/2,n 1], n>2, n n/2 nn/2(> 1), n n 322-3 a 1 <a 2 < <a n n, 3 322-4 [0, 1] a, b, c a b + c +1 + b c + a +1 + c +(1 a)(1 b)(1 c) 1 a + b +1, a + b + c 323 323-1 x, y f (1) f(x, x) =x (2) f(x, y) =f(y, x) (3) f(x, y) =f(x, x + y), f gcd

168 323-2,, A A (, ), P, Q P, Q,, ( ) 100 Z, Z 100 A, A A A A A X, A A X A A A A,,, 100, 100 100, 100 100 100, 100

33 169 323-3 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 33 331 331-1,??,,

170?, MathLetter proposal MathLetter 59(19963, 4) 331-2 3, a, b, c a P 60 3d, e d, eb, c B, C, PBC a A ABC, P, 332

33 171 332-1 AB C AB, AC, CB k 1, k 2, k 3 k 2k 3 k 2, k 3 D, E C AB k 1 F FDCE 1 D, C, E FDCE F DECF M MC = MD = ME M CDE DCE DE ADC, CEB AC, CB, ADBE, F k 1 F DCE F F, F, F F C k 1, CF CF F F DCF = CDE, F DCE F DCECD DCF = CDE, DCF DCF, F F

172 2 F ( 161 ) 332-2 ABCD, BD < AC EF ABCD, BCAD, LM ACBD LM EF = 1 AC 2 BD BD AC 34 341 341-1 2 1999 4sin 2 kπ 2 2000 3 k=0 1 20001999 2 n f(n) = 4sin 2 kπ 2 n+1 3 k=0, n? (n 1) 2 n f(n) = ( 1) 2n +1 3 4sin 2 kπ 2 n+1 = = 2 n k=0 2 n k=0 k=0 3 2 1 cos kπ 2 n 1+2cos kπ 2 n

34 173 n 2 (1) n =1 : 2 f(1) = 1+2cos kπ 2 k=0 = (1 + 2 cos 0) 1+2cos π (1 + 2 cos π) 2 = (1 + 2)(1 + 0)(1 2) = (1 2 2 )=3 (2) n =2 : 4 f(2) = 1+2cos kπ 4 k=0 = (1 + 2) 1+2cos π (1 + 0) 1+2cos 3π (1 2) 4 4 = f(1) (1 + 2)(1 2)=f(1)(1 2) = f(1) = 3 (3) n =3 : f(3) = 8 k=0 1+2cos kπ 8 = (1 + 2) 1+2cos π (1 + 2) 1+2cos 3π (1 + 0) 8 8 1+2cos 5π (1 2) 1+2cos 7π (1 2) 8 8 = f(2) 1+2cos π 1+2cos 3π 1 2cos 3π 8 8 8 = f(2) 1 2 2 cos 2 π 1 2 2 cos 2 3π 8 8 = f(2) 1 2 1+cos π 1 2 4 = f(2) ( 1) 2 1+2cos π 4, (2) 1+cos 3π 4 1+2cos 3π 4 f(3) = f(2) f(2) f(1) = f(2) 1 2cos π 8 f(n +1) = f(n) f(2) = = f(n) f(n 1) f(1) = 1 ( )

174 ( ) f(n +1) f(n) = = = 2 n+1 l=0 0<l<2 n+1 l: 0<l<2 n l: 1+2cos lπ 2 n+1 1+2cos 1+2cos 2 n lπ 2 n+1 lπ 2 n+1 k=0 1+2cos kπ 2 n 2 n <l<2 n+1 l: lπ 1+2cos 2 n+1 (**) 2 n <l<2 n+1 l: lπ 1+2cos 2 n+1 = = 0<l<2 n l: 0<l<2 n l: 1+2cos (2n+1 l)π 2 n+1 1 2cos lπ 2 n+1, f(n +1) f(n) = = = 0<l<2 n l: 0<l<2 n l: 0<l<2 n l: 1+2cos 1 2 2 cos 2 lπ 2 n+1 1 2 = ( 1) 2n 1 0<l<2 n l: lπ 2 n+1 1 2cos 1+cos lπ 2 n 1+2cos lπ 2 n lπ 2 n+1 n 2 ( ), f(n +1) f(n) = f(n) f(n 1) (n 2),( ) f(n) f(n 1) f(n) = f(2) f(n 1) f(n 2) f(1) f(1) = ( 1) n 1 3 (n 2), f(1999) = ( 1) 1998 3=3

34 175 341-2 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 341-3 1, 2, 3,, 2000 ( ) 2000 342 342-1 (+) ( ) 4, 4 4 4,

176 ( ),,, ( ), ( ) ( ) 4 4 0, 1, 2, 3, 4,2 2 16, 2 0,1 ( ), (+, +) : 0 (, ) :2 (+, ) :1 (, +) : 1, ( ) 1( ) 0( ), 2 2 01 3 3, 4 4 2 2 4 4 2 2, ( ), 0 1 ( ), ( ), 4 4 0, 1, 2, 3, 4 4 4 5

34 177 342-2 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 ) a 1 a 2 a n b 1 b 2 b n := a 1 b 1 z 1 + a 2 b 2 z 2 + + a n b n z n z 1 z 2 z n a 1 a 2 a n a 1 a 2 a n b 1 b 2 b n b 1 b 2 b n z 1 z 2 z n z 1 z2 zn ( ),, a i a 1 <a 2 < <a n, c j >c k (j<k) j k, a j a k b j z j b k z k a j <a k c j >c k

178 a j < a k < c j > c k > a j a k < c j ḳ > c, 2 a j c j a k c ḳ a j < ḳ c >, <, (b i a k c j ) (c i ),, (z i ), ( ) 35 351 351-1 n n 1 k k 2 k=0 k

35 179 351-2 99 C 1,C 2,,C 99, n S n : ( P ) C ic j (1 i, j 99), C i C j S, C j C i S 99m(> 0) nm, n(m) C 1,,C m P n(m) S, C 1 = {x 1 },,C m = {x m } S n(m) <m 1 S C i, C i P, n(m) m 1 n(m) =m 1 m m =1, 2 m m +1 C 1,C 2 a, a a (a- )m 1, (ā- )m 2 (m 1 + m 2 = m +1,m 1,m 2 > 0) (1) a- ā- a, (2) a- P m 1 1 S 1, (3) ā- P m 2 1 S 2, S = {a} S 1 S 2 S 1+(m 1 1) + (m 2 1) = (m +1) 1,(m +1) 1,

180 351-3 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),, b i = a i /x i [ ] (b 1 + a 2 )(b 2 + a 3 ) (b n 1 + a n )(b n + a 1 ) (b 1 + a 1 ) (b n + a n ), 0<a 1 a n,0<b k b n (k =1,,n) (b j + a k )(b n + a j ) (b j + a j )(b n + a k ) (1 j<k n) b j b n, a j a k - (j, k) =(1, 2), (2, 3),,(n 1,n) (b 1 + a 2 )(b n + a 1 ) (b 1 + a 1 )(b n + a 2 ) (b 2 + a 3 )(b n + a 2 ) (b 2 + a 2 )(b n + a 3 ) (b n 1 + a n )(b n + a n 1 ) (b n 1 + a n 1 )(b n + a n ) (b n + a 2 ) (b n + a n 1 ) 352 352-1 (a, b, c, d), a, b, c, d a>b>c>d>1, a! d! =b! c!

35 181 352-2 (1999KMO ) N, f : N N?, (1) f(2000) = 1999, (2) n, f(n),f(2n),f(3n),,f(kn),, (3) f(f(n)) = 1 11 (3) (2),(1) p, q2 t f(pq t )=q, f(2000) = 1999 f(n) =1 352-3 ABCD [ ] (1) A B C D ABCD 1/9 (2)

182

4 4-1 A B A B A B? 4-2 n a 1,a 2,,a n f(x) =a 1 sin x + a 2 sin 2x + + a n sin nx x f(x) sin x a 1 +2a 2 + + na n 1 4-3 (1) i =1, 2, 3, P i = x i(i 1) <x 2 i(i +1),x 2, i = j P i P j = ( )

184 (2), f f(m, n) = 1 (m + n 2)(m + n 1) + m 2 f(m, n) =f(p, q) m = p, n = q 4-4 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

5 1 11,, 11 5, 6, 6, 5 5 6, 6 6 (1) 11 = 462 6 5 6 (2), 7 (3), (1)

186,, 10 10 5, = 252 5,,,, 2 A, B, C 25 A B C 2, A A A, C, B?, A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 a 0 a 1 a 2 a 3 a 4 a 5 a 6 a 7 0,1 (A, B, C) = (0, 1, 1) a 3A c B C a 1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 =25 (1) a 2 + a 3 =2(a 1 + a 3 + a 5 + a 7 ) (2) a 4 =(a 5 + a 6 + a 7 )+1 (3) a 4 = a 1 + a 2 (4) a 1 1 (5) a 2 a 5 + a 7 a 5 + a 7 = b

187 a i, (2)a 2 =2a 1 + a 3 +2b (1)a 2 1 (4) a 4(1) a 2 2 (3) a 6 + b (1) a 4 3 3, 2, 1 (1) a 1 + a 2 + a 3 +2a 4 =26 3a 1 +3a 2 + a 3 =26 9a 1 +4a 3 +6b =26 (1 ) a 1 (5) a 1 1, (1 ) 9 3 > 26 a 1 < 3 a 1 =2 4a 3 +6b =8 (a 3,b)=(2, 0), a 2 =2a 1 + a 3 +2b =2 2+ 2+2 0=6,,, 3 4 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 1k 3 C k 2k 4,,

188 5 A, B, C n>ν ν An 2 + Bn + C<n! ν An 2 + Bn + C<n 3 n! ν> A, B, C, 3 ν = A + B + C +3, 6 1 2,, 7 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,, : p, q 8 K 1, K 2 AB ABK 1 K 2 K 1 P T- (, ), CDAB CPD K 1 C, D

189 9 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 f(n) =n 10 nn 3 2n 1 E E k,, E n k k k 11 Q + x, y f(xf(y)) = f(x) y f : Q + Q +, 12 A, B (1) 1 A, (2) A 2 k +2(k =0, 1, 2,),

190 (3) B 2 k +2(k =0, 1, 2,), 1988, 1989, 1990 AB, 13 n>1, a 1,a 2,,a nd (i) a k a k 1 = d for k =2, 3,,n, (ii) a i = b mi i for i =1, 2,,n m i (> 1) b i n =3 (a 1,a 2,a 3 )=(1, 25, 49) = (1 2, 5 2, 7 2 ) d =24 (1 2, 5 2, 7 2 ) (r 2, (5r) 2, (7r) 2,w) w (7r) 2 =(7r) 2 (5r) 2 w =73r 2 r =73 (a 1,,a n )=(b m 1 1,,bm n n ) n, n +1 1 rm,,b m n n r m,w)=((b 1 r m/m 1 ) m 1,,(b n r m/m n ) m n,w) (b m1, mm i m i r m n, w b mn n r m = b mn n r m b m n 1 n 1 rm, w =(2b mn n b m n 1 n 1 )rm, r =2b m n bmn 1, n n 1 n>1

191 14 3 a,(a 2 1)(b 2 1) = c 2 1 2 (b, c) 15 f f(x) : x, f(x +1)=f(x)+1, 3 )=(f(x)) 3 n f(x + n) =f(x)+n ( ), 1 (y 3 ) (f ),? x, f NQ + p q Q+ (, p, q N) p 3 p 3 f q + q2 = f q 3 +3p2 +3pq 3 + q 6 p 3 = f +3p 2 +3pq 3 + q 6 q p 3 3 3 p p f q + q2 = f q + q2 = f + q 2 q 3 2 p p p = f +3 f q 2 +3f q 4 + q 6 q q q

192 t = p p 3 3 p q ( ), f q 3 = f, q p(p + q 3 )=q 2 t 2 + q 4 t (qt p)(qt + p + q 3 )=0 qt + p + q 3 > 0 t = p p q, f q = p q 16 a m + a 1 a n + a 2 1 a m, n 3 1 x m + x 1=q(x)(x n + x 2 1) + r(x) q(x), r(x), x r(x) N(x n 1 + + x +1)<Mx n 1 N, M m, n (Nr(x), M = nn ) x, x>m x>m x n + x 2 1 >x n >Mx n 1 > r(x) r(x), x n + x 2 1 r(x) 0, (x n + x 2 1) (x m + x 1) 2 m n x m + x 1=x m n (x n + x 2 1) + (1 x)(x m n+1 + x m n 1) (1 x)x n + x 2 1 (x n + x 2 1) (x m n+1 + x m n 1)

193 m n +1 n m 2n +1 0 (x m n+1 + x m n 1) x m 2n+1 (x n + x 2 1) = x m n x m 2n+3 + x m 2n+1 1 (x n + x 2 1) (x m n x m 2n+3 + x m 2n+1 1) f(x) =x n + x 2 1, g(x) =x m n x m 2n+3 + x m 2n+1 1 m n m 2n +3, m 2n +1 0, m =5,n =3( ) g(x) < 0 for all x (0, 1) f(0) = 1, f(1) = 1 f(x) (0, 1),(m, n) =(5, 3), x 5 + x 1 x 3 + x 2 1 = x2 x +1 17 n 3,Γ 1, Γ 2,,Γ n 1 O 1,O 2,,O n 1 (n 1)π O i O j 4 1 i<j n O i Γ j 2x ij sin x ij = 1, x ij > 1 O i O j O i O j x ij = x ji, 1 i<j n x ij (n 1)π 4 ( ) O i (O i ) H k O 1 O k, O k+1,,o n H

194 (1) H O i (i>k), Γ j 2x ij 2x ij, 4x ij, 4x ij < 2π (k <i n) j(=i) 1 2 2x ij < (n k)π (1) i>k j=i (2) H O i (i k) O i Γ j 2x ij, O i H y i 2x ij <y i + x i(i 1) + x i(i+1) (1 i k, x 10 = x 1k ) j(=i)

195 H yi =(k 2)π 2x ij < (k 2)π + i k j=i 1 i k 2x i(i 1) 2x i(i 1) (1) 4x i(i 1) 2π 1 i k, 2x ij < (k 1)π (2) i k j=i (1), (2) 4x ij = 2x ij < (n 1)π i<j i=j ( ),,, 18 n n, n n,,, 19 x i? n x1 + + x n x 1! x 2! x n!! n (, [y]y )

196 x i 0 n x i 1 f(n) =ln(n!) f(n +1)=f(n)+ln(n +1) ln(n +1) (1,f(1) (2,f(2) (n, f(n) F F Jensen F (x 1 )+ + F (x n ) n x1 + + x n F n x i F (x i )=f(x i )=ln(x i!), F (x) F ([x]) = f([x]) = ln([x]!), x1 + + x n ln(x 1!) + +ln(x n!) n ln! n ln (F x i ),,, 2 (A) n =2 t n =1 ( ) n =2 x 1, x 2 m x 1 =[m]+d + e, x 2 =[m] d (e =0 1), x 1! x 2! ([m]!) 2 ([m]+d + e)! [m]! [m]! ([m] d)! ( d = e =0,x 1 = x 2 ) n = k n =2k n =2 x 1! x 2k!=(x 1!x 2!)(x 3!x 4!) (x 2k 1 x 2k )! 2 2 x1 + x 2 x2k 1 + x 2k!! =(1) 2 2 x i, x 2i 1 + x 2i,

197 x 1 + x 2, 2 x1 + x 2 x3 + x 4 x2k 1 + x 2k (1) =!!! 2 2 2 x1 + x 2 =! x 2 3 + x 4 + x 2k! x2k 1! 2 2 2 x1 +x 2 2 + x 3 +x 4 2 + + x 2k 2k 1+x 2k 2! =(2) k,[x + n] =[x]+n, n = x n (x, n ) x1 +x 2 + +x 2k 2k 2k 2 x1 + + x 2k (2) =! =! k 2k n =2k (1) x 2i 1 = x 2i (2) x 2i 1 + x 2i x i (B) n = k +1 n = k m = 1 k (x 1 + + x k ) x k+1 =[m], n = k +1 k+1 x1 + + x k +[m] x 1! x k! x k+1!=x 1! x k![m]!! =(3) k +1 (3) = [x] k+1 [km + m]! =([m]!) k+1 k +1 x 1! x k! ([m]!) k n = k x i (A), (B) - (backward induction) n, x i,,, 20 a, b, c 1 a + 1 b + 1 c =1 a + bc + b + ca + c + ab abc + a + b + c

198 1 a = x, 1 b = y, 1 c = z xyz, x + y + z =1 yz + x + zx + y + xy + z 1+ xy + yz + zx yz + x x + yz, xy + z z + xy, zx + y y + zx z =1 x y xy + z = xy x y +1= (1 x)(1 y) = (z + y)(z + x) (z + yx ) 2 = z + xy,, 21 N = {0, 1, 2,} x, y N xf(y)+yf(x) =(x + y)f(x 2 + y 2 ) f : N N 1 f(x) c xc + yc =(x + y)c f f(y) f(x) > 0 f(y) f(x), (x + y)f(y) >xf(y)+yf(x) > (x + y)f(x), (x + y) f(y) >f(x 2 + y 2 ) >f(x) f(x 2 + y 2 ) f(x) > 0 f(y) f(x), f 2 [ 1] g(x) f(x) f(0) g(0) = 0, g(x) f(0) xg(y)+yg(x) =(x + y)g(x 2 + y 2 ) f

199 [ 2] y =0 x>0 g(x 2 )=0 g(1) = g(4) = 0, x = y =1 g(2) = 0, (x, y), x 2 + y 2 = z 2 z (x, y) g(y) = y x g(x) ( ) (x, y) =(3, 4) g(3) = 0, g(1) = g(2) = g(3) = g(4) = 0 [ 3] x =2n>4 x =2n +1> 4, (x, y) =(2n, n 2 1), (x, y) =(2n +1, 2n 2 +2n) y>x, x>4 ( ) y(> x) [ 4] g g(x 0 ) = 0x 0 > 4 ( ) g(x i+1 )= (x i+1 /x i )g(x i ) x 0 <x 1 <x 2 < g(x i+1 ) > g(x i ), g g(x i ) < f(0) x i g(x) f(0) [ 5], g, f 1 f 3 f f(0) = k, S = {x N : f(x) =k} y =0 f(x 2 )=k, x 2 S (1), x 2 + y 2 = z 2 x S y S (2),2 2n =(2 n ) 2 S, x = y =2 n 2 2n+1 S, n 2 n S (3) n p(n) n p(n),( ) [ ] x 1 > 1, x 12 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

200 x 12 n = x 1,x 2,,x r p(x r )=2 x r2, (3) x r S (2) x 1 S x 2 S x r S, n = x 1 S SN