005 9 45 IMO () (,00074), l,b A l C ( C A B ), IMO 4 AC l D, DE a 0, a, a, E, B E AC B E a a + - a +, 0, a 0 a l F,AF G( G A)? :G AB CF f : Q{ -,}, O ABC, B < x y, xy C, AO BC D, ABD x + y {0,},f ( x) f ( y) -? ACD EF BA CA, G H AG AC,AH AB : EFGH 4 IMO 5 a b c > 0,ab + bc + ca : a + 6 b + b + 6c + c + 6 a abc 6 f : R R, f ( x + y + f ( xy) ) ( f ( x + y) ), x y 7 a, a,, a, >, g A,A,,A A k a + a + + a k k, k,,,, G A, A,, A : G A + g G +, IMO l,ab ACB - ABC 60 4 IMO 5 5 A A A, B, B,, B - : () i i -,B i A i A i + ; () i, i -, S i A A i + A A i,b i A i S i A i + A i A i + : A B A + A B A + + A B - A 80 6 P :P, P 4 7 ABC,X BC, C B X ABX ACX P Q : PQ X 8 ABCD, AD BC E, C B E
4 AC BD F M CD, N ABM M, c, AN, BN AM BM :E F N a c a p + k F k - + F k + + c { a } IMO 4, { a } a a + c,a - 0, a - a - c,, a 0 a c, c, 0, 0, a > 0, 0,,,, a + + a, a + > a + ; a + g ( x), x Q, x > 0 ; a + - a, a + < a + f ( x) - g ( - x), x Q, x < 0 ;,, x 0,{ a }, x y, a + a + - a, a + a + - a +,, a + - a < 0 a p + a p - a p - a p + k + a p + k - a p + k - x a b > 0, y b a > 0, a b, a > b, ( a, b) k, a p > a p +, a p + < a p + < < a p + k - < a p + k, a, b, ( a mod b), a p + k > a p + k +, ( b, a) a p, a p + b, a, b, ( a mod b), a p + j F j - + F j, j,,, k, F 0 0, F, F i + F i + + F i, a p + k F k - + F k, a p + k + a p + k - a p + k - ( F k - + F k ) - ( F k - + F k - ) F k - + F k -, k ;, k, a p + k +, p, a ap + x a b, a b ( a, b) ( u mod v) u v, r 0 a, r b, r ( a mod b),, r i + ( r i - mod r i ),, r, r + 0 r ( x) ( x) x,g ( x) ( - ), { -,} f : Q { -,} : f ( x) x + y 0, x > 0, y < 0 f ( x) g ( x),f ( y) - g ( - y) - g ( x),f ( x) f ( y) - xy,x y a > b, r ( b mod a) b,, x y, ( y) ( x) +, g ( y) - g ( x),f ( y) - f ( x), f ( x) f ( y) -, x y x + y,x y, x a b > 0, a b
005 9 5, y b - a b y < 0, f ( y) - g ( - y) - g a - b b ( a, b) a, b, ( a mod b), ; ( a - b, b) a - b, b, ( ( a - b) mod b), a > b, ( ( a - b) mod b) ( a mod b), b, g a f ( x) f ( y) - g a - b b, y > 0, 0 < x < < y <, a < b a < b, b - a < b,, ( a, b) a, b, a, ( b mod a),, ( b - a, b) b - a, b, b - a, ( b mod ( b - a) ), ( b mod ( b - a) ) a,a < b - a, r a, r 4 ( ( b - a) mod a) ( b mod a), y b - a, b, b - a, a, ( b mod a),, ( y) ( x) +,f ( x) f ( y) - 4 IMO 5 u + v + w u + v + w, u v w u a + 6 b + a + 6 b, v b + 6 c + b + 6 c, w c + 6 a a + 6 b + b + 6 c + c + 6 a ab + bc + ca abc a + b - ab c + 6 ( a + b + c) ab - ( ab) abc, c + 6 a, b + c - bc a c + a - ca b, ab + bc + ca abc bc - ( bc) abc ca - ( ca) abc + 6 ( a + b + c) + [ ( a + b) + ( b + c) + ( c + a) ] abc abc {4 - [ ( ab) + ( bc) + ( ca) ]} [ ( ab) + ( bc) + ( ca) ] ( ab + bc + ca) ab + bc + ca + 6 ( a + b + c) abc abc, abc abc, a b c 7 a b c ( ab) ( bc) ( ca) ab + bc + ca,,, 7 a b c 6 z x + y, t xy, z 4 t g ( t) f ( t) - t f ( z + g ( t) ) ( f ( z) ), t z R, z 4 t g (0) c, t 0, f ( z + c) ( f ( z) ), z R, f ( x) 0, x c,c < 0,f (0) 0, c g (0) f (0) 0,, c 0 g, t,g ( t) g (0) c,f ( t) t + c, c 0,f ( x) x g,, x, f, m, [,]
6 g ( u) - g ( v), u v m, y > x R m,y - x [,], u v m, y - x g ( u) - g ( v),x + g ( u) y + g ( v) x 4 m 4 u, y 4 m 4 v, ( f ( x) ) ( f ( y) ), m,,, z R, s - <, m c 4,, x y 0, c z + g ( s) f ( x) f ( y) f ( z + g ( t) ), f ( x) [4 m, + ),, [,] f ( x) [4 m, + ),f ( x) [ m, + ) g, a b, g ( a) - g ( b) d > 0 u max{ a, b, c} k,v u + d, u + g ( a) v + g ( b), u 4 a, v 4 b, v > u c f ( u) f ( v), g ( u) - g ( v) ( f ( u) - u) - ( f ( v) - v) ( v - u) ( v - u ) v + u ( g ( a) - g ( b) ) v + u d u + u + d h ( u) u + -, X < -, -, d, [ k, u + d f :x X, f ( x) - ;x X, + ), (0, h ( k) ] h ( k), l > k, h ( l) t [,], u [ k, l ], h ( u) t, t g ( u) - g ( v), k u < v u + d, u [ k, l ] f ( - z) f ( z) z, ( -, - ], f, g ( s) f ( s) - s - - s, s - ( -, - ]g ( s) f ( z + g ( t) ) ( f ( z) ) ( f ( z) ), z R, f ( z) 0, f ( x) 0,c f (0) ( c 0) s,f ( s) - z R, z s - g ( s) 4s, ( f ( z) ) f ( z + g ( s) ) f ( s) -,, s - g ( s) < 0, 0 s - g ( s) < 4s, g ( s) - - s,, s < - s >, c, x,f ( x) 0, f -, - f ( x), f, f ( x) x, f ( x) 0 7,A 0 0, k,,,, u + d < u + d, m l + u v m,,m d, > 0 R, x, f ( x),z,,, 0 a k A k ka k - ( k - ) A k - A k k - ( k - ) A k - A k x, x k A k - A k G A A A A A, k,,,, x x x -,
005 9 7 g G k, G A + g G a k A k x x x - + [ k - ( k - ) x k ] k [ k - ( k - ) x k ] k,,, x x x - + ( + ) x ( + ) x x x - x + ( k - ) x k k + ( k - ) x k, k x k, k,,,,,e ABD,F ACD, [ k - ( k - ) x k ] k k + - k ( k - ) x k, [ k - ( k - ) x k ] k - ( k - ) x k,x k,, FD AB k,,,, ED AC, G A + g G +, a a a IMO,CF H AB l, GH l AB l, X, AXF AEF 90,A F X E, EFD EAB FED, DF DE DF DE DC DA,, DCF DFA, AFD FCD ACH AGH GH l, ABC, ACB, < ABC,O ABC OAC 90 -, ADB DAC + ACD 90 - +> 90 ABD, ACD O E AB F O ACD ABC, FAC 90 - ADC ADB - 90 -, FDC 90 - OAC DA GH P, PAH OAC 90 -, PHA ABC (ABC AHG),, APH 90,AD GH EF AD,,AD EF, GH EF AGH DFE, HE GF AD, HE GF AD EF GH AD,, EFGH,, AGH DFE, DF AG F ACD,AG AC,, AC AG DF AF CF ACF, DF AG ACF AC,, ACF FAC 60, - 60 4 IMO 5 5 ABCD AB
8 CD, AC BD S, M BC, BSC BC N ( ) AMD AND : A DM N, AD BC, X XA XB a, XC XD b, BN CN BS CS AB CD XA XD a b BN + CN, CN a + b b BN + CN BC a - b,, CN ( a - b) b, XN ab a + b a + b XM a + b,, XM XN ab XA XD A D M N C i A i A i +, i,,, - < i < -, A A i A i + A A A A i A i +,,, A B i A A C i A, B i C i, A B A + A B A + + A B - A A C A + A C A + + A C - A A A A,,ax + by - ( a + b) k i, i,,, -, A C i A A + - i C A + - i A C i A A + - i C A + - i, i,,,, A C A + A C A + + A C - A A C A + A C A - + + A C A 80, S KLT P 4 P P 4, 6 P KL ABC s t ( P s t, P S, s t ),S T P s KL BC, t, ST P,, P K L P, KL ST,KL ST LSB CBS 4 LBS K L P, Z K s L s ZK s X, ZL t Y Z ST s t A B, P SXZYT, S S KLT 4 S SXZYT AZ a, B Z b, K X Y S AB k x y d K L KU LV s t,ab U V a + b KL UZ + ZV ak x + bk y S S KLT ( a + b) ( d - k), 4 S SXZYT S g SABT - S AXZ - S BYZ ( a + b) d - ax - by, ax + by - ( a + b) k 0 ( a + b) xy k ( ay + bx), ab( x - y) ay + bx, 7 0 L K ABC AB AC, ABC BC CA D E KL DE ABC : 4, AB BC,
005 9 9,LB LS AM BM,M N ABM LA LB,, S AB,ASB 90 AX BX X,X DE ABC T, ABC AB, AB I B T AB BC, T,N, C E, DEC 90 -, AIB 90 + C AM BM, N,M N T DE, AIT + AET 80,A I T E I E AT, C AIT 90 - AET, A I T E AEI 90,, ATI 90 ASB ATB,S T,KL DE ABC 5, ABX ACX BX D F, AX EG, DE FG,DE FG AXB 5 PQ BX AX M N, MD MPMQ MF, NE NP NQ NG,MD MF, NE NG, PQ DE FG, AB AC AX, m, m BC ABX,DE m ABX U, FG m ACX AMD + BMC V PQ DE FG,,, PQ UV W U V X,, W X 8 ABM, AN N AN M, BN AM BM AB AB E AB F EF P Q,, N PQ N PQ S E F P S Q 6 PQ,PQ AB 6 APB E AQB F ABCD,, APE AB E ABC 80 - ADC, AQP AQF AB F ABD ACD, APQ ADC AS AM,, AS AM PQ DC, B PQ BCD, BS BM PQ CD, AS BS AM BM AS P AMD, BS P BMC,, ASB AS P + BS P 80 - AMB,S ABM,S M N M, S M,N S ABM,, BN AM BM, AS BS AM BM, N, S N,N PQ
45 IMO (), m,d, a 4, 6 a si, b si, c si,a b c (0,) a + b + c, a - a a ( - a ) a + - a + - a, b - b 9, c - c 9 a b c - a + - b + - c a a - a + b b - b + c c - c 9 (,00074) ( ) k, 0 00, : ( 4, ),, ( ( 4), ) k, q : () ( ) 4, ;, (), 0,c, ;, (), m c + m, 5 N, m {,,, N}, k,, k,, + ( N) N N, A (B ), ( ), k ( a + b + c ) ta si - si a - a, ta si - si b - b, ta si - si c - c,,ta + ta + ta : ( ) 994-007 Chia Academic Joural Electroic Publishig House All rights reserved http://wwwckiet
005 0 () N 004 A B?, k, k k, () N ( N > 004), N, 004, k, 6 m m A,X i,f (,) {,} k,, i, Y j j,, i j m X, X,, X m, Y, Y,, Y m, A (, ),, i ( i,,, ), i 004 004,, -, {,,, },, - 7 IMO,, 8 G,f ( G) G k,, F(,) {}, F(,) {},, g ( G) G 4 F(,) {} c,, k, G, ( g ( G) ) c ( f ( G) ) 4 k, k k k k,,, k, i i, i,,,,k, 0 00,, ( F(,) {,}, F (,) {,}, a, R, S), a R S F(,) {,}, a R, R S >, F(,) {,}, F( i, i + ) { i} ( i,,, - ), F( -, ) { -, - }, a S, (), R ( i < j ( i, j), F ( i, j) { }, a, R, S),, ( a, S) k,,, i -, i, i ( i,,, - ), -, -, - R, R, i i ( i ) R -, () R, k, R ( R - ) R k +, - k,,,,, - M,,,,, -, k, R ( R - ), i i ( i,,, - ) R M R ( R - ) k R M i ( i,,, - ), i R ( R - ), R M C R, () i ( i,,, - ),i k + R M C R C k ( - ),k - R M, i i i ( i,,, k), i 0 00, k 5 000 i ( k + ), k +,,,, k, -,F( i, j) i j -,F( i, j), 994-007 Chia Academic Joural Electroic Publishig House All rights reserved http://wwwckiet
4,, 4 c,, i j ;, ; a ij, -, i j > ; ; V 0, V, V V 0,, G, G 4, c ABCD, AB, H :, () G,H ;, - (), G,H, (),, G X Y, AB,X, -,, Y H AB, H X Y AD DC CB, AB,, H (), H, V V M, V V BC CD DA H,A C V S M 4 M, V, B D G H V V, AB V V,, S M > 4, M,G K,K,, ( ),, H M, H,, N c, c,, c, M -, -,, N -,,, -,,, N : 4, K, S N ( 4) 4, M N, A, A,, A -, X, Y, Z, 4, S M > 4, S N >, M N 4 Z, X Y K ( M N 4, ), A i XYA j A i A j, i < j - ; XYZA i XA i, i - ; YXZA j YA j, j -, ( a ij ) i j : c, c,, c S M N S M + S N - S M N > 4 M N, 4, -, 994-007 Chia Academic Joural Electroic Publishig House All rights reserved http://wwwckiet
005 0 5-4,,,, N A ( ) 5 N B N (i), I k, I k -, 0, (ii), (iii), (ii), (iii), N k N, N k -, N k - 4, {,,, N}, a a a 5 0,, (ii) (iii),, N B N B ( ( ) N a k a k - a N,a k N 0 0, N a k, N a k a k -,, N k - a k a k - a, N k a k a k - a N, I j { N N j - < Nj }, j,,, k I k,, 0 00000000000 048,, 048 B, 004 6 j,,, k, I j - 004 004, I j, : {,,, },X, X,, X 004, Y, Y,, Y I j, j,,, k, 004 {,,, } (i) I j, ;,4 008 -, (ii) I j, ; 004 004, S {,,,} (iii) I j A { X, X,, X 004, Y, Y,, Y 004 }, I k,n, I k X { X, X,, X 004 }, Y { Y, Y,, Y 004 } (i) ;N, I k (ii) S 4 096,, j, I j 4 096-4 008 88 I j (i), I j -, I j A i j, +,, i j 004,X i, I j - N j - ( ) Y j g I j ( ) i j, X i Y j 5 ( B, I j - I j,, I j - B ) X i X j (i) S - X I j (ii), I j -, i Y j 7 8 A ( X I j,, I j - i Y j ),,,I j - (iii) I j (iii), I j - X, X,, X 004, I j,, I j -, I j - + N j -, I j,, I j - I j N j - N j - I j - : N j -, I j - (ii) ; N j -,I j - (i) I j (i) k mi X i, i, X i k, S (i), I {} ), : N B N 0 () 004 004 0000,, 004 A () N > 004 N, A 88, Y, Y,, Y, Y j, Y j i ( i ), Xi Y j 6, X i Y j,, X i X i Y j + X i Y j - Y j 4, i i ( i ), Xi 4 i 004 - X i k X ( k 994-007 Chia Academic Joural Electroic Publishig House All rights reserved http://wwwckiet
), Y V i x i, V i x i, ( X i ) k 4, S - X i 0,,, C 0 8 + C 8 + C 8 + C 8 9 > 88, k 5,S - X i K 4, 5 C 0 7 + C 7 + C 7 + C 7 + C 4 7 99 > 88, 6, k 6,X 6 S 88 A,, C 0 + C + C + C + C 4 + C 5-88 498 S 6 Y,y i C 6, X ( Y E,, Y i < E,C x i X ),, 498 996 S X 4 096-996 < 004,, : A,A m A m - A m - A m - B m -, m,,, X m -, Y m - X, X,, X m -, X { m + }, X { m + }, { m + } ; Y, Y,, Y m -, Y { m + }, Y { m + }, { m + }, m,a m m - + j, j m -, ( g ( G) ) A m, ( f ( G) ) 4 {,,, c m + }, ( m -, m ], 0 < 004 <,, 004, +, 004, {,,,} 7 IMO 8 G V, V,, V,E, E E f ( G) E x i V i A i, A i x i y i z i V i K i x i E, i y i f ( G), i z i 4 g ( G) E i A i K, V i, y i E i A i x i,, E i y i x i x i, E i y i E i C x i E i x i E x i i f ( G) E E, ( f ( G) ) 9 E B m - m - m -,, m + m ( m ), A m A i E i G i {,,, m + } m m, i,,,, : A m ( f ( G i ) ),A A m - 9 E i, f ( G,A m - X, i ) A i E i K K X,, X m -, Y, Y, Y m -, V i K {,,, m + }, A m 4,,z i f ( G i ) z i ( f ( G i ) ) ( f ( G i ) ) ( f ( G i ) ) 9 ( f ( G) ) 9 Ei yi i 4 g ( G) ( f ( G) ) 9 ( g ( K ) ) ( f ( K ) ) 4 (C4 ) (C ) 4 f ( G), (!) 4 [ ( - ) ( - ) ( - ) ] 4 (!) (4!) [ ( - ) ( - ) ] 4 (4!),, c 994-007 Chia Academic Joural Electroic Publishig House All rights reserved http://wwwckiet C x i
8 45 IMO () (,00074) N + f : m N +, ( m + ) ( ) f ( m) + f ( ) : : a, N +,f ( ) ( a) 4 k, m 4 k - 5 : a b, N + { x } : x 0 a, x b, ( ) ( k, ), N +, k ( k, ) k m () : 5 IMO 6 m, ( m) ( m)( ) ; 6 ( > ),P () : a N +, x, ( x) ax ; () a N +, ( x) ax x + x + + x, 0,, x x - >,P 7 p, OR OQsi 00 si 40 5 si 00 si 80 si 40 5 si 40 47 (cm) l A ( x, y ), B ( x, y ) () l,ab x, x x 4 5 4 5 - + y, 4 y 4 5, y - A 4 5, 4 5 4 5, B, AOB 4 5, - 4 5 () l k ( k 0),A B ( x - ) 4 y k x - + y, 4 5 5 (4 k + ) x - 0 (8 k + 5) x + 64 k 0 x + x 4 (8 k + 5) 5 (4 k + ), x 64 k x 5 (4 k + ) x 5 y + 4 k, 5 k 5 (4 k + ) y - 60 ky - 64 k 0 k y + y 5(4k + ), y 64k y - 5(4 k + ) k OA k OB y y -,, AOB x x () () AOB ( )
005 9 p k (mod m), k k (mod ), k k {,,, : m},k k (mod m),k k, k {,,, m}, p + ( r, s),( k, m) ( r, m), ( k, ) ( s, ),, ( a), a a ( a), k k ( k) a, : ( m)( ) p 5, k ( k) p p - () p, p, ( k, ) p l, p l k, [, p ] p l, d, p l +, l 0,,, d -, p - l - p - l -, [, ] k (, ] k ( ) k p 5, k ( k) p p - k, k p p - k p s, p -, p s, p s (+ )( s) p p - p -,s p( s), p s, p +, p p - (+ ) p s ( s) s s s p 5, p +, (+ ) < p - p + ( ),, s <, p,ps ( p + )( s), p ( s),, p ( s) s, s s s s ( s) ( p + ) p, p s 4 ( p + ) p p 5, () m, ( )( l) (+ ) - ( l) k N +,( k, m) ( k, m) ( k, ), l, ( l) m m ( m) ( k, m) ( k, m) ( k, ),,( l) l ( l) k k > l ( l > ), l ( l), + - k {,,, m}, a -,x a - ( x) ax ( r, s) r k (mod m), s k (mod ), r m, s, r m, s ( r, s) m k m ( m) ( k, m) ( k, ) k ( r, m) ( s, ) r m r s m ( r, m) ( s, ) s - ( ) ( p ) p + p l ( p - l - p - l - ) l 0 (+ ) p - p - a N +,p, a -, ( a - ) a a -, x a - ( x) ax () p,( p p ) p p +, p a N + p,x a p - p p ( x) ax,() ( a p - p p ) ( a p - )( p p ) a a p p - p p + a a p - p p p,,x a p - p p x a -,( x) ax, a, 0,,,, ( x) x x,x l, 0, l () + l x ( x) ( l),( x) ax a, 0,,, m, f () +
0 f () ( + ) 4 t + t 4,f () + f (),,f () + f (),f () m, ( f ( ) + ) ( + ),,, ( f ( m) + ) ( m + ), m X f ( ), >, - X k, f ( k) k, X N + f ( k) k, - k, k + f ( ) f ( k) + f ( ) ( k + ) ( k + ) [ ( k + f ( ) ) + ( - f ( ) ) ] A ( k + f ( ) ) + ( - f ( ) ),, A,A, 4, ( - f ( ) ) k + f ( ) k k >, X k,, ( - f ( ) ) 0, A k, N +,f ( ) X,, p, ( f ( p - ) + ) p,f ( p - ) + p f ( p - ) + p f ( p - ) + p, ( p - ) + [ ( p - ) + ] p >, ( p - ) + > ( p - ) ( p + ), [ ( p - ) + ) ] [ ( p - ) + ( p - ) ] ( p - ) p,,f ( p - ) + p,,a k, A m X,P (mod ) p -, f ( p - ) p - 4 a, b k + k - 4 k 5 (mod m),, b 4 k + k - 4 k + (mod m), 4 b 4 k + 4 k + 4 k + 6 4 b + 4(mod m) m,, b b + (mod m) ( b, m) ( k + k -,4 k - 5) ( k + k -, k + ) (, k + ),, ( b, m), 0,x b (mod m) 0,,, X, x x - + x - b - + b - - b - ( b + ) b b b (mod m), ( x, m) ( b, m) 5 IMO 6 6,P > X x (mod ) {,,, - },X, x X, x X,x x X, X,x X,x A {, x } X A,x A A { x, x x } {, x, x, x x } x k +, A k A k + A k { xx k + x A k } A k { xx k + x A k },, A k + < X, k +,, A k + X,, m, X A m A, A k ( k > ) X ab, a >, b >,a b, x y x (mod a), x - (mod b), y - (mod a), y (mod b) x y, X x y < ab x (mod ), y (mod ),, x y a >, b > >, x y X, k ( k > ), k - k - +, X 4,, X
005 p k, p, k x - x +,x (mod ), x (mod ) x - (mod ), X -, p k, p, k > 0,, 4, p k p k, X -, p, k,p -, X,P (mod ) 7 A B,AB bd ), ac p ( ef - bd ) p p k AB,p k + 8AB, p ac, ( AB) k S, S S abc 4 R ( a b c A i A i, R ), (), A 8 ABC A 8 AB BC + BC CA + CA AB - AB 4 - BC 4 - CA 4 A,A,,A 6, 6 S, AB BC CA ( S) p A B C :m mi{(ab),(bc),( CA)} p m ( S),, p m S ( AB) +( BC) +( CA) m + ( AB),( BC),( CA), ( AB) +( BC) +( CA) m +, m + m + m 0,( AB) ( BC) ( CA) 0, :, P Q ( BC) ( DA) ( AC) ( BD), p a b c d e f, AB a, CD c, BC bp, DA dp,ac ep, BD fp ABCD, ac p ( ef - bd ) ac p ( ef - bd ), ( ef - bd ) ( ef - bd ) ef + bd - bdef,, bdef, ( ef - p A, A,,A 8 ( A i A j ) +,,, p A A A,, A A A A 8, ( AB),( BC),( CA),, 0 (),,, 4, : 4, A A A A 4 A 5 A 6 A 7 A 8 ; 8, A A A A 4 A 5 A 6 A 7 A 8,A A A 5 A 7, (), A A A A A A A 4 A A A A 4 A 4 A A A p A A 4 A 4 A, ( A A ) ( A A 4 ) ( A 4 A ) 0 ( PQ) + ( A A ) 0 A A A S, A B S p ( A A ) + C D,A B C D,( A, AB A ) +, S p, CD,( AB) ( CD) 0