第四届全国大学生数学竞赛预赛试题评分标准

1 15 z 3x2 4y2 1 x y z t t x y z 5 tz 3x2 4y2 t2 1 t 10 z2 4 3x2 4y2 0 15 2 2 15 P P L A L A L P L y x2 P x0 y0 1 P y0 x2 0 2 L y k x x0 y0 L x x2 k x x0 y0 x1 0 f x 0 x 0 R 0 1 f x f x dx R 0 1 f x dx f x 0 0 Z N 0 1 f x dx Z N 0 1 f x f x dx Z N 0 f x f x f x f x dx 1 1 lim N Z N 0 f x f x f x f x dx lim N Z N 0 f x f x 2 dx lim N 1 f x N 0 1 f 0 8 Z 0 1 f x dx Z 0 1 f x f x dx 1 f 0 10 4 10 A B C n P t At2 Bt C f t detP t t detP t P t f t Re 0 b 0 c 0 b b2 4ac 2a 6 b2 4ac 0 b2 4ac b Re b b2 4ac 2a 0 8 b2 4ac 0 b2 4ac i 4ac b2 Re b 2a 0 2 5 10 1 x n 1 x 3 P i 0aix i x 1 n Pn 1 i 0 ai Pn 1 i 0 ai 1 x n 1 x 3 1 1 x xn 1 2 1 x n 1 x 4 2 1 x n 1 x 4 n X i 0 1 iCi n2 n i 1 x i 4 2 xn 1 2n 1 x 4 n2n 1 1 x 3 n n 1 2 2n 2 1 x 2 n n 1 n 2 6 2n 3 1 x 1 xn 1 6 2n 3 1 x 1 n2n 1 2 1 x 1 n n 1 2 2n 2 1 x 1 n n 1 n 2 6 2n 3 1 x 1 xn 1 2n 3 n 2 n 1 n n2n 1 2 n 1 n n n 1 2 2n 2n n n 1 n 2 6 2n 3 n 1 X i 0 ai n n 2 n 7 3 2n 4 10 2 6 15 f 0 1 R f 0 f 1 R1 0 f x dx 0 f x 6 1 x 0 1 n Pn 1 k 0f k n 1 2 f 0 f 1 c 0 1 f c 0 2 f x 6 1 f x 1 n n X k 1 g k n 1 1 n n 1 X k 0 g k n Z 1 0 g x dx 1 2 12 n 1 X k 0 f k n n 1 X k 0 g k n n 1 2 1 2 2 15 3 7 25 A 2 2 2a B 4 b 31 1 AX B BY A a 6 2 b 4 3 2 A B a 3 b 2 3 3 A B a 2 b 3 1 AX B B A BY A A B 2 A B 2 24b 2a31 2 24b 0a 2 11 b B A a 6 2 6 B A B A 4 b22 312a 4 b22 01 3b 41 2a 3 2 A B b 4 3 AX B BY A a 6 2 b 4 3 10 2 A B trA trB A B a 3 b 2 3 12 a 3 b 2 3 A 2 2 23 B 4 2 3 31 A B 2 5 2 2 5 2 0 A B A B 15 3 A A B B b 3 16 B g x1 x2 4x2 1 6x1x2 x 2 2 3x1 x2 2 5x2 1 y1 3x1 x2 y2 x1 g x1 x2 y2 1 5y22 18 B 1 19 A f x1 x2 2x2 1 4x1x2 ax 2 2