17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,

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1 17 ( ) II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ OA OB () PQ () OR : RP (4) PQR a a α β x x + a A(α) O() B(β) (1) AOB 18 a () AOB a 4 a a 1 f(x) x f(x) ( x + ax (1) f(x) ) f(t) dt 1 e x () f(x) x 1 a () () a f(x) 1

2 5 n f(n) (n + 1)(n + )(n + 5)(n + 7)(n + 9) (1) k f(k) f(k 1) a(k + 1)(k + )(k + 5)(k + 7) a m () m (k + 1)(k + )(k + 5)(k + 7) f(m) k1 6 a b f(x) x ax + bx + x, (1) a b f(x) () y f (x) + 7 x 7 n 5x + y n (1) n x y () x y ( ) a b 8 a b c d A A + BA 4E c d AB + B 1E ( ) B a + d ad bc 1 E 1 9 a n, b n (n 1,, ) tan(a n ) 1 n, < a n < π tan(b n ) 1 n + 1, < b n < π (1) < x < π tan x x () tan(a 1 + a + + a n ) () a 1 + a + + a n + b n (4) lim n (a 1 + a + + a n ) 1 a y ax y ax y ax 1 V

3 1 (1) a 1 1, a a 1 + 4, a a + 5 8, a 4 a a () a n+1 a n + n + 1 n+1 n+1 a n+1 n a n n a b f(n) an + b, f(n + 1) f(n) n + 1 a(n + 1) + b (an + b) n + 1 an + a b n + 1 a a b 1 a b f(n) n 1 n+1 a n+1 f(n + 1) { n a n f(n)} { n a n f(n)} a 1 f(1) 6 n a n f(n) 6 n 1 a n + f(n) n n + n

4 4 (1) PQ OQ OP 1 OB OA + OB ( OA + OB) R O Q () (1) A P 1 PQ PQ 1 4 OA + OB 1 OA 4 + OA OB + OB B cos () OPB AQ OR RP PA AB BQ QO 1 OR RP OR : RP 4 : (4) OPB 1 4 OAB OPQ 1 OPB PQR 7 OPQ PQR OAB (1) x x + a x 1 ± 1 a ( ) AOB 18 AOB 18 αβ < αβ a a < () AOB ( ) (a > 1) α 1 + a 1 i β 1 a 1 i OA OB a, AB a 1 AB OA + OB OA OB cos AOB ( a 1 ) ( a) + ( a) a a cos 4(a 1) a + a a a 4( )

5 5 4 (1) k f(t) dt f(x) (x + akx 1)e x k (t + akt 1)e t dt [ {(t + akt 1) (t + ak) + } ] 1 e t [ {t + (ak )t (ak 1) } ] 1 e t ak 1 (a 1)k 1 a 1 k 1 a 1 ( f(x) x + a ) a 1 x 1 e x g(x)e x dx {g(x) g (x) + g (x) g (x) + }e x + C () (1) ( f (x) x + a ) ( e x + a 1 ( x + a a 1 x + 1 a 1 f (1) 1 + a a a 1 f (x) (x + )(x 1)e x x + ) e x a ) a 1 x 1 e x a 1 x 1 f (x) f(x) x 1 a 1 () () f(x) x 1 f (x) + + f(x) 5e e { 5e` (x ) e (x 1)

6 6 5 (1) f(k) f(k 1) (k + 1)(k + )(k + 5)(k + 7)(k + 9) (k 1)(k + 1)(k + )(k + 5)(k + 7) (k + 1)(k + )(k + 5)(k + 7){(k + 9) (k 1)} 1(k + 1)(k + )(k + 5)(k + 7) a 1 () (1) m (k + 1)(k + )(k + 5)(k + 7) 1 1 k1 m {f(k) f(k 1)} k1 1 {f(m) f()} 1 1 {f(m) 945} 1 6 (1) f(x) x ax + bx + f (x) x ax + b f (x) + a, b a, b 4 f(x) x x 4x + f (x) x 4x 4 (x + )(x ) f(x) x f (x) + + f(x) { 11 (x 7 5 (x ) () f (x) + 7 x 4x 5 ( x + 1 ) ( x 5 ) S 5 ( S x + 1 ) ( x 5 ) dx ( ) 4

7 7 7 (1) 5x + y n 1 1 (x, y) (n, n) 5n + ( n) n 1 5(x n) + (y + n) 5(n x) (y + n) 5 k n x k, y + n 5k x n k, y n + 5k ( ) n x k y + 5k x y k >, + 5k > 6 < k < 1 k 7, 8, 9 (x, y) (9, 5), (6, 1), (, 15) () ( ) (x, y) (n k, n + 5k) (k )

8 8 8 A + BA 4E (A + B)A 4E AB + B 1E (A + B)B 1E A + B (A + B) 1 A 4(A + B) 1, B 1(A + B) 1 B A A + BA 4E A E 1 A A (a + d)a + (ad bc)e O 1 (a + d)a (ad bc + 1)E i) a + d ad bc + 1 ii) a + d A ke (k ) 1 k E E k ±1 A ±E A E a + d ad bc 1 A E a + d ad bc 1 (a + d, ad bc) (, 1), (, 1), (, 1)

9 9 9 (1) f(x) tan x x < x < π f (x) 1 cos x 1 tan x > f(x) f() < x < π, f(x) > tan x > x () tan a n 1 n tan a 1 1 tan a 1 8 tan a 1 18 tan(a 1 + a ) tan a 1 + tan a 1 tan a 1 tan a tan(a 1 + a + a ) tan(a 1 + a ) + tan a 1 tan(a 1 + a ) tan a tan(a 1 + a + + a n ) n n + 1 (i) n 1 ( ) (ii) n k tan(a 1 + a + + a k ) ( ) k k + 1 tan(a 1 + a + + a k + a k+1 ) tan(a 1 + a + + a k ) + tan a k+1 1 tan(a 1 + a + + a k ) tan a k+1 k k (k + 1) k(k + 1) + (k + 1) 1 k 1 (k + 1) k + 1 k (k + 1) (k + k + 1)(k + 1) (k + k + 1)(k + ) k + 1 k + n k + 1 ( ) 4 (i) (ii) n ( ) tan(a 1 + a + + a n ) n n + 1

10 1 () () tan(a 1 + a + + a n + b n ) tan(a 1 + a + + a n ) + tan b n 1 tan(a 1 + a + + a n ) tan b n n n n n 1 n + 1 n + 1 n + n + 1 n + n < a n < tan a n < b n < tan b n n > 1 < n a k + b n < k1 n tan a k + tan b n k1 n 1 k + 1 n + 1 k1 < n 1 k n ( 1 k 1 1 ) k k k(k 1) + 1 n n + 1 n + 1 < n + 1 n 1 < a 1 + a + + a n + b n < 1 < π 1 a 1 + a + + a n + b n π 4 (4) () a 1 + a + + a n π 4 b n < b n < tan b n 1 n + 1 lim b n n lim n!1 (a 1 + a + + a n ) π 4

11 11 1 C : y ax l : y ax C y P(x, ax ) l Q(x, ax) P l a PH C l PH : HQ : PQ 1 : a : 1 + a t OH h PH ax ax H Q P PQ ax ax ax(1 x) O x 1 x h ax(1 x) 1 + a, HQ a x(1 x) 1 + a, t OQ HQ 1 + a x a x(1 x) 1 + a x + a x 1 + a dt dx 1 + a x 1 + a t 1 + a x 1 V 1+a 1 V π h a x (1 x) dt π 1 + a x 1 + a dx 1 + a πa {x (1 x) + a x (1 x) }dx (1 + a ) x (1 x) dx 1 x (1 x) dx 1 6 ( πa 1 V (1 + a ) + a 1 ) πa a

12 1 m n I(m, n) t m (1 t) n dt I(m, n) 1 (t m+1 ) (1 t) n dt m + 1 [ ] 1 1 m + 1 tm+1 (1 t) n n I(m + 1, n 1) m + 1 n m + 1 n 1 m + 1 m!n! (m + n)! t m+n dt β α + n m + 1 I(m + n, ) m + n m!n! (m + n + 1)! (x α) m (β x) n dx t m+1 (1 t) n 1 dt dx x α + (β α)t dt β α x α β t 1 β (x α) m (β x) n dx (β α) m+n+1 t m (1 t) n dt α (x α) m (β x) n dx m!n! (β α)m+n+1 (m + n + 1)! x (1 x) dx!! 5! (1 )5 1 x (1 x) dx!! 6! (1 )6 1 6

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

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