第三届全国大学生数学竞赛决赛试卷参考解答

2012 15 A 1 0 1 B 1 1 2 C 1 1 2 D 3 1 0 E 3 1 2 E A B C AD ABC n AB AC 0 1 1 0 1 3 2 0 0 l a b c l n 0 l AD 0 l 0 c c c 6 0 l 0 1 1 x 3 0 y 1 1 z 2 1 15 f x a b f x a b f x f a f a x a Z x a x t f t dt x a b 1 x a b Newton Leibniz f x f a Z x a f u du f a Z x a f a du Z x a du Z u a f t dt f a f a x a Z x a dt Z x t f t du f a f a x a Z x a x t f t dt 2 x a b Z x a x t f t dt x t f t t x t a Z x a f t dt f a x a f x f a f x f a f a x a Z x a x t f t dt 10 k0 k1 kn A1 A2 An f x sink0 x A1sink1x Ansinknx 0 2 A1 A2 An 1 A1 A2 An 0 f x 0 2 2k0 A1 A2 An f x 0 2 2k0 F1 x 1 k2 0 sink0 x n X i 1 Aik2 0 k2 i sinkix F1 0 F1 2 0 F 1 x f x F1 x 0 2 N Rolle F 1 x 0 2 N F 1 x 0 2 N 1 F 1 x 0 2 N F0 x f x Fs x 1 s k2s 0 sink0 x n X i 1 Aik2s 0 k2s i sinkix F s 1 x Fs x s 0 1 2 Fs x 0 2 N f x 0 2 N g x sink0 x n X i 1 Aik2s 0 k2s i sinkix Fs x 1 s k2s 0 g x k0 k1 kn s n X i 1 Ai k2s 0 k2s i 2 2 n X i 1 Aik2s 0 k2s i sinkix 6 n X i 1 Ai k2s 0 k2s i 2 2 n X i 1 Ai k2s 0 k2s i 0 g m 4 k0 6 2 2 n X i 1 Ai k2s 0 k2s i 2 2 n X i 1 Ai k2s 0 k2s i 0 2 g m 4 k0 6 2 2 n X i 1 Ai k2s 0 k2s i 2k0 f x 2k0 10 an lim n an 1 lim n an lim n n a 1a2 an 1 lim n a1 a2 an n 1 xn lnan lim n xn 0 lim n xn 0 A 0 xn6 A 0 6 x 6 ln2 ex6 1 2x n Sn i Z 1 6 i 6 n xi6 ln2 Tn i Z 1 6 i 6 n xi ln2 1 n n X k 1 xk 1 n X k Tn xk Tn n ln2 0 lim n Tn n 0 1 n n X k 1 exk 1 n X k Sn exk 1 n X k Tn exk 6 1 n X k Sn 1 2xk Tn n eA 6 1 Tn n 1 eA 2 n n X k 1 xk 1 n n X k 1 exk e 1 n n P k 1 xk 3 lim n 1 n n X k 1 ak lim n 1 n n X k 1 exk 1 zn x n xn 0 n 1 2 lim n zn 0 yn xn zn yn 0 lim n yn 0 lim n yn 0 xn6 yn e 1 n n P k 1 xk 6 1 n n X k 1 exk6 1 n n X k 1 eyk lim n 1 n n X k 1 ak lim n 1 n n X k 1 exk 1 15 A B 3 2 2 3 AB 80 4 3 2 9 6 201 BA AB 9 2 AB BA BA 9 BA 2 C CBA BAC I2 AB 9 A BA 9I2 B ABAB 9AB AB AB 9I3 0 BA 9I2 CB A BA 9I2 B AC 0 BA 9I2 90 09 4 20 Ai i I Bi i I F F F i I P P 1AiP Bi i I Q Ai i I Bi i I R Q i I Mn R Ui R T Mn R AiT TBi Mn Q Ui Q T Mn Q AiT TBi Mn R Mn Q R Q n UR i I Ui R UQ i I Ui Q UR6 0 n2 UR UQ Ui R Ui Q Ui R Ui Q AiT TBi T1 T2 Tl f t1 t2 tl det l X k 1 tkTk UR6 0 s1 s2 sl f s1 s2 sl 6 0 f Q r1 r2 rl f r1 r2 rl 6 0 P l X k 1 rkTk UQ i I P 1AiP Bi 15 F x G x 0 lim x x F x G x 0 i 0 lim x Z xF xt costdt 0 ii lim n Z 0 F t G t cos t ndt 0 lim x 0 Z 0 F t G t cos xt dt 0 i lim x xF x 0 0 Z 2 xF xt costdt 6 xF x Z cost dt 0 x 5 min n 2 o max n 2 o lim x Z 2 xF xt costdt 0 Dirichlet Z 2 xF xt costdt X k 0 Z 2 k 1 2 k xF xt costdt x X k 0 1 k Z3 2 2 F x t k costdt Z 2 xF xt costdt 6 x Z3 2 2 F xt costdt 6 2xF x 2 0 x lim x Z 2 xF xt costdt 0 ii x 0 lim x Z 0 x F xt G xt costdt 0 i lim x Z 0 0 x F xt G xt costdt 0 0 06 2 lim n Z 0 0 n G nt G n 1 t costdt 0 Z 0 0 n G nt G n 1 t costdt Z n 0 0 G u cos u ndu n n 1 Z n 1 0 0 G u cos u n 1du 1 n 1 Z n 0 0 G u cos u ndu n n 1 Z n 0 0 G u cos u n cos u n 1 du n n 1 Z n 1 0 n 0 G u cos u n 1du I1 I2 I3 6 lim x xG x 0 lim x G x 0 I1 1 n 1 Z n 0 0 G u cos u ndu 6 Z 0 0 G nu cosudu Z1 n 0 G nu cosudu Z 0 1 n G nu cosudu 6 G 0 n G n Z 0 1 n cosudu 0 n I2 n n 1 n X k 1 Z k 0 k 1 0 G u cos u n 1 cos u n du 6 n n 1 n X k 1 G k 1 0 Z k 0 k 1 0 u n n 1 du Lagrange 0 n 1 2 n X k 1 k 1 0G k 1 0 2 0 2 n 1 2 n X k 1 G k 1 0 6 n 0 n 1 2 1 n n X k 1 k 1 0G k 1 0 n 2 0G 0 2 n 1 2 0 n I3 n Z 0 n 0 n 1 G n 1 u cosudu 6 nG n 0 Z 0 n 0 n 1 cosudu 6 n 0G n 0 n 1 0 n lim n Z 0 0 n G nt G n 1 t costdt 0 I1 lim x Z 0 0 F xt G xt costdt 0 lim x x Z 0 0 F xt G xt costdt 0 x x