pn k C ứ 1,2, ,n : k n k p k n ! n k 0 n p k n! : nCnk11 kCnk , pn k Cnk pn k k 0 n n n n n n 1 k 0 k 0 k 0 k 0 k 0 k pn k k.Cnk pnk 0 n.Cnk11 pnk n Cnk11 pnk n Cnk1 pn1k n 1 n pn 1 k n n 1! n ! k 0 x1 , x2 , , x2 n C i 1, 2, , 2n 1 : xi xi 1 n C ứ : n n +1), (2, n + ) 1, 2, , 2n n n, 2n) A, B y : A B f : A B x1 , x2 , , xk 1 , xk , xk 1 , , x2 n A k 2n x2n f x , x , , x xk k 1 , xk , xk 1 , , x2 n x1 , x2 , , xk 1 , xk , x2 n , x2 n 1 , , xk 1 B f : A B : n 3 C k 0 3k n n ứ P x 1 x Cni xi n i 0 l 0 nê'u k : 2k k ' k 3 nêu n 1 ứ n 3 P 1 P P Cni i 2i 3 Cn3i i 0 i 0 P 1 1 1 n n n 1 n n P 1 i i cos i sin 2 2 3 P n n 1 n n 1 i i cos i sin 2 2 3 n n 3 C k 0 3k n 2n cos n n Cn n 1 ứ 0 nê'u k : 2k k ' k 3 nêu ứ P x x3 x x5 x n Cn P(x) 6n 2n P x ak x k Cn a3k k 0 k 0 P 1 , P P 6n 2n : P 1 P P ak k k 3 a3k k 0 n 4n 2n Cn a3k k 0 n C k n k 0 S k 0 cos kx n n Cnk cos kx , T Cnk sin kx k 0 k 0 n n : S iT Cnk cos kx i sin kx Cnk eikx eix k 0 k 0 n 1 cos x i sin x n n x x x x nx nx 2cos cos i sin 2cos cos i sin 2 2 2 2 n n x nx S cos cos 2 6: (VMO_1996) C k n k a1 , a2 , , ak kn a1 , a2 , , ak n : i) t 1, 2, , k : st as at ii) s 1, 2, , k : as s 7: (VMO_2009) C n : n a, b a b 1, n ) ứ : f(n), f(n – ) ứ ct f(n) f(n) ứ (VMO 1977) n P(n) n ứn+ n ứn+ n n n P(n + 1) = P(n) + 2n P(n) = P(n – 1) + 2(n – ) = C = P( ) + [ + + a1 , a2 , , an n i 1,2, , n 1 Sn i– –i ứ f : T S n1 a1 , a2 , , an1 , n 1,2, ,n > + n, 1 i n 1 : a1 , a2 , , an n– ứ 1, 2, , n 1 ứ ai1 , ai2 , , an n, 1 i n 1 Cni 11 a1 , a2 , , an a1 , a2 , , an an = n Sn1 a1 , a2 , , an an = n T T + n – 1)] = + (n – 1)n a1 , a2 , , ai1 , n : a1 , a2 , , an1 f n 1 Sn Sn1 Cni 11 Sn1 2n1 S2 2n1 2n2 22 n i 1 = 2n1 2n 2 2 n n n 1 C p : Sn Sn p j j n, Sn 1, 2, , n j Sn f ( n) Sn Sn Sn Sn a) C ứ : g (n) nf n 1 , n b) C ứ : f n n 1 f n f n 1 , n c) C ứ : f n g n 1 n Sn Sn : 1,2, ,n g(n) p : Sn Sn a) j Sn j C n : S n \ j S n \ j ) f(n – g (n) nf n 1 , n r : Sn Sn r(1) = j j b) n– ) nj : TH1: r j T ứ r T f n 2 Sn \ 1, j Sn \ 1, j S : S f n 2 :S T : s S , s : Sn \ 1, j Sn \ 1, j , ( s) r : S n S n r 1 j r j r i s i , i 1, j : T S f n 2 TH2: r j Y ứ r Y f n 2 Sn \ 1 Sn \ 1 X : X f n 1 : X Y : s X , s : Sn \ 1 Sn \ 1 , ( s) r : S n S n x r 1 j nêu ' s i j r i 1, ' s i j r i s i , nêu : r j s j j ) Y X f n 1 f n n 1 f n f n 1 , n c) ) ) : f n 1 g n 1 n f n f n 1 n 1 f n nf n 1 f n g n f n f(1) – g(1) = – = –1, f(2) – g(2) = – = n : f n g n 1 II ứ n C n S 1, 2, , n n S 1, 2, , n ứ C ứ C n S 1, 2, , n ) S S ứ S C n (n ) m (m ) ứ pn k n k p k n! k 0 n 1,2, ,n k C ứ : f : A B a) f a1 , a2 A, f (a1 ) f (a2 ) a1 b B, a A : b) f f c) f 2) Cho A, B : ) f : A B ) f : A B ) f : A B 3*) Cho A, B ) f : A B ) f : A B )C ứ : )C n i 1 n Ai Ai i 1 1i j n Ai Aj 1i j k n a2 a1 , a2 A, a1 a2 f (a1 ) f (a2 ) f (a) b b B, !a A : f (a ) b | A| | B | | A| | B | | A | | B | : f f nh Ai Aj Ak (1) n A1 A2 An N 99 N N T f : T T N T T f : N a1 a2 a2010 : f ( N ) b1b2 b2010 bi , i 1, 2010 a) : N = a1 a2 a2010 T : {1, 2,3, 4,5, 6, 7,8} nên bi {1, 2,3, 4,5, 6, 7,8}, i 1, 2010 N 99 N f ( N ) 9999 999 99 nên f ( N ) 99 2010 ch sô ' f(N) T f b) N1 x1 x2 x2010 , N y1 y2 y2010 T cho f ( N1 ) a1a2 a2010 , f ( N ) b1b2 b2010 f ( N1 ) f ( N ) bi , i 1, 2010 xi yi xi yi , i 1, 2010 N1 N c) ) N b1b2 b2010 T C ứ ) f C PT P a1 a2 a2010 f(P) = N ứ bi ) N f ( N ) 9999 999, N T 2010 ch sô ' : 2 N N f ( N ) T 9999 999 N T N T 2010 ch sô ' N f N (vì f song ánh ) N T NT N 9999 999 102010 2010 ch sô ' N T N : T 2 C A = {1, n A ) C A D A n a) f : C D b) A C2nn X MC : X Y n A C Y : f : C D : M ( X M ) (Y \ M ) : X M Y M M ( X M ) (Y \ M ) X M Y \ M ( X M ) (Y \ M ) ) = X M Y Y M Y n ( X M ) (Y \ M ) D C cho f ( M ) f ( N ) ứ M, N : ( X M ) (Y \ M ) ( X N ) (Y \ N ) (1) X M, X N Y \ M ,Y \ N ) : X M X N X M X N M X M Y M X N Y N N Y \ M Y \ N Y M Y N M X N Y \ N ND M M : X N : Y \N Y Y N n Y N N Y N X N f ( M ) ( X M ) (Y \ M ) X N Y N N MC C D C2nn f N i) ii) n n ) : N N – ) C S S X XT [ m(X) T m m( X ) X T T C : C ứ C n r < n – r + Cho X r n aX r ứ ĐS Cnr r 1 m : a) b) ĐS a) Cnmm11 n n m b) C m 1 n 1