ORIGINAL TEXT I II A B 1

4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2

281 I II A B 292 3

I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2 2x + y 1 (3) (x + y)(y + z)(z + x) + xyz (4) x 3 + y 3 3xy + 1 1 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2yz + 2zx, x 3 + y 3 + z 3 3xyz = (x + y + z)(x 2 + y 2 + z 2 xy yz zx) 4

(5) x 6 1 (6) x 3 + (a 2)x 2 (2a + 3)x 3a 2 (7) (a b) 3 + (b c) 3 + (c a) 3 2 [ ]x 3 + 2(a + 1)x 2 + (a 2 + a 3)x a 2 3a 5

2 (1) 1 1+ 2+ 3 (2) 1 2+ 3+ 5 (3) 5 + 2 6 (4) 7 4 3 (5) 15 6 6 (6) 4 + 15 6

(7) 7 3 5 (8) 9 + 4 4 + 2 3 (9) 2 + 3 4 3 0 < a < 3 3 a 2 + 2 a 2 + 4a + 4 2 a 2 6a + 9 3 3 [ ]a < 0 1 2a 2 a 2 a 7

4 k = 0, 1, 2, 3 a k (a b)(a c) + b k (b c)(b a) + c k (c a)(c b) 8

5 x + y + z = 3, 1 x + 1 y + 1 z = 1 3 (1) (x 3)(y 3)(z 3) (2) x 3 + y 3 + z 3 9

6 i x = 3 + 3i x 3 2x 2 + 4x 6 4 7 x, y, z x + y + z = 1 x + 1 y + 1 z = xy + yz + zx x, y, z β 2 4β+8 4 [ ]α = 1 + 3i, β = 1 3i ( α n+2 α n+1 +2α n +4α n 1 +α 3 2α 2 +5α 2 )3 10

8 A, B, C, D x 3 = A + B(x 1) + C(x 1)(x 2) + D(x 1)(x 2)(x 3) 11

9 x, y, z x + y + z = 0 x 3 + y 3 + z 3 = 3 x 5 + y 5 + z 5 = 15 x 2 + y 2 + z 2 5 5 [ ] x, y, z x + y + z = 0, x 3 + y 3 + z 3 = 1, x 4 + y 4 + z 4 = 2 x 2 + y 2 + z 2 12

I II A B c P (x) x α P (α) P (x) = (x α)q(x) + C (1) P (α) = C (2) P (x) x α P (α) = 0 C = 0 1 (1) x 3 + 2x + 1 (x 2) (2) x 4 7x 2 + 12 (x + 1) 2 (1) x 3 + 6x 2 + 11x + 6 (2) 3x 3 + 5x 2 + 5x + 2 13

3 x 4 + ax 3 13x 2 + bx + 24 (x + 2)(x 1) a, b 4 x P (x) x + 2 4 x 3 6 P (x) (x + 2)(x 3) 14

5 x P (x) x 2 3x + 2 x + 4 x 2 4x + 3 3x P (x) x 2 5x + 6 6 P (x) = x 4 + ax 3 + bx 2 + c x 1 5 x 2 + 1 2x + 1 a, b, c 15

7 x 5 1 6 (1) x 2 1 (2) x 2 + 1 (3) (x 1) 2 6 P (x) (x α) 2 P (α) = 0 P (α) = 0 16

8 x P (x) (x 1) 2 2x 3 x 2 P (x) (x 1) 2 (x 2) 9 (x 100 + 1) 100 + (x 2 + 1) 100 + 1 x 2 + x + 1 7 7 ω 3 = 1, ω 2 + ω + 1 = 0 17

10 (x + 2)(x 1) 2 (x 2 + x + 2) x 3 1 11 f(x) = x 4 ax 3 bx 2 cx + d x 2 + x + 1 x 2 + 1 10 a, b, c, d 8 8 [ ]x 2 x + 1 = 0 α f(α) = (α 1) 100 + α 100 + 1 18

12 4x 3 4x 2 + 1 9 13 x 2 + 3x + 1 = 0 9 x f(x) = 0 ± c a 19

14 f(x) = x 4 + ax 3 + bx 2 + cx + 1 x f(x) = 0 a, b, c 10 10 [ ] 3 28 27 + 1 3 28 27 1 (1) α (2) α 20

I II A B c 1 (1) a 2 x + 1 = ax + a (2) { xy + x + y = 1 x 2 + y 2 + xy = 3 21

x + y + z = 2 (3) xyz = 2 1 x + 1 y + 1 z = 1 2 (4) x + y + z = 2 x 2 + y 2 + z 2 = 6 x 3 + y 3 + z 3 = 8 22

2 x 2 + kx + 3 = 0, x 2 + x + 3k = 0 k 3 x 3 + 6x + 20 = 0, x 2 2x + a = 0 (1) a (2) a 23

4 x, y 4x 2y = kx, 3x + 3y = ky x = 0, y = 0 k 5 x 2 + 2x + 3 = 0 α, β (α 3 + α 2 + 1)(β 3 + β 2 + 1) 11 12 11 [ ]x 2 + 2x + 3 = 0 α, β α 5 + β 5 12 [ ] x 199 + 10x 5 = 0 199 199 24

6 x 3 + kx 2 + kx + 1 = 0 k 7 x 4 + ax 2 + 4 = 0 a 25

8 x 2 + ax + 1 = 0 x 4 + (a + 1)x 3 + (a + b + 1)x 2 + (ab + a + b)x + a 2 + b 2 1 = 0 a, b 26

I II A B c 1 (1) (x + 1)(x 3) < = x + 2 (2) (x 2)(x 1) x 3 > = 0 27

(3) x 2 + 3x + 7 > = x 1 + x 2 (4) x 2 2x 3 > = 2 x 2 28

(5) x 2 (a + 1)x + a < 0 (6) ax 2 x a 2 x + a > 0 a > 0 29

2 (1) x 2 xy + y 2 > = 0 (2) x 2 + y 2 + z 2 xy yz zx > = 0 30

3 x 2 + 2y 2 + 2xy + 2x 2y + 8 4 x + 2y + 3z = 1 x 2 + y 2 + z 2 31

a + b 2 >= ab a = b 13 a + b > = 2 ab a > 0 b > 0 a + b + c 3 > = 3 abc a + b + c + d > 4 = 4 abcd 13 50 50 50 10 90 50 30 ( ) > = ( ) ( ) = ( ) 32

5 x, y x + y > 2 = xy 6 x, y, z x + y + z 3 > = 3 xyz 14 14 [ ]n x, y, z x n + y n + z n = xyz (1) n = 1 (x, y, z) x < = y < = z (2) n = 3 (x, y, z) 33

7 x, y, z (5) (6) (1) x + 1 x > = 2 (2) (x + 1 x )(y + 1 y ) > = 4 (3) (x + 4 x )(y + 9 y ) > = 24 (4) ( y x + x y )( y x + 4x y ) > = 9 34

(5) (x + 1 x )(y + 4 y )(z + 9 z ) > = (6) (x + 4 x )(x + 9 x ) > = x + 4 x > = 2 x 4 x = 4 x + 9 x > = 2 x 9 x = 6 (x + 4 x )(x + 9 x ) > = 24 (x + 4 x )(x + 9 x ) = x2 + 36 x + 13 > 2 = 2 x 2 36 x + 13 = 25 2 (x + 4 x )(x + 9 x ) > = 25 35

8 x 2 + 2x + 1 x 2 x + 1 15 9 x 4 2x 3 + x 2 + 8 x 2 x + 1 16 15 [ ] 2 x + 3 = 1 (0 < x, 0 < y) xy y [ ] t 0 < t < 1 O(0, 0), A(0, 1), B(1, 0), C(t, 0) AB D ACO = BCD t ACD 36

10 (a + b + c)(ab + bc + ca) > = 9abc a > b > c > 0 37

11 (1) a 2 + 1 a 2 > = a + 1 a a > 0 (2) a 3 + 1 a > 3 = a2 + 1 a > 0 a2 38

12 a < 1, b < 1, c < 1 (1) ab + 1 > a + b (2) abc + 2 > a + b + c 39

13 a, b, c a < b < c, a + b + c = 0 1 2 < = a2 + b 2 + c 2 (c a) 2 < 2 3 40

14 a, b a b (1) ( a+b 2 )2 a, 2 +b 2 2 y O x (2) ( a+2b 3 ) 2 a, 2 +2b 2 3 y O x 41

(3) s + t = 1 sa 2 + tb 2, (sa + tb) 2 y O x (4) log( a+b 2 ), log a+log b 2 y O x 42

15 a, b, x, y a+b = 1 n (ax+by) n < = ax n + by n 16 x 2 + y 2 + z 2 > = tx(y z) x, y, z t 43

I II A B c 1 f(x) = ax 2 + bx + c (1) (2, 1) (6, 3) (2) (3, 3), (1, 1) y = x 44

(3) g(x) = x 2 + 2x (3, 0), (5, 0) (4) (2, 9) x 12 45

(5) 1 (0, 3), (3, 0) 46

2 f(x) = ax 2 + bx + c a, b, c, b 2 4ac y O x y O x 47

3 x 2 2ax + a + 12 = 0 a (1) (2) (3) 48

4 f(x) = 0 x 2 (1) f(x) = 0 1 (2) f(x) = 0 1 (3) f(x) = 0 (4) f(x) = 0 1 < x < 3 (5) f(x) = 0 1 < x < 3 (6) f(x) = 0 1 < x < 3 (7) f(x) = 0 (8) f(x) = 0 1 < x < 2 2 < x < 3 (9) f(x) = 0 1 < x < 2 5 < x < 6 49

5 ax 2 + 2bx a + 2 = 0 x > 0 a > 0 50

6 y = tx t 2 t 0 < = t < = 1 xy 17 17 [ ]C 1 : x 2 +y 2 = 2 C 2 : (x a) 2 +(y 1) 2 = 2a 2 +2 l a 1 < = a < = 1 l 51

7 m, n 2x 2 2(m 1)x + n 2 = 0 0 < x < 2 m, n 52

8 f(x) = x 2 + ax + b A(1, 1), B(2, 3) (a, b) 53

I II A B c 1 a > = 0 f(x) = x 2 2x {x a < = x < = a + 1} g(a) (1) g(a) a (2) b = g(a) 54

2 0 < = x < = 2 y = x 2 2ax + 1 (1) a < = 0 (2) 0 < a < 1 (3) a = 1 (4) 1 < a < 2 (5) 2 < = a 55

3 f(x) = x + a, g(x) = x 2 x + 2 a (1) f(x) < g(x) x (2) f(x) < g(x) x (3) f(x) > g(x) x (4) f(x) > g(x) x 56

4 (1) a x 7a 2 3a + 1 + 6ax x 2 m(a) (2) a 3a 2 2a 1 < = 0 m(a) 57

5 x 3y > = 6, x + 2y > = 4, 3x + y < = 12 x + y x 2 + y 2 18 18 [ ] P (x, y) 4x + y < = 9, x + 2y > = 4, 2x 3y > = 6 2x + y, x 2 + y 2 58

6 (1) x + y = 1 x 2 + y 2 (2) x 2 + y 2 = 1 x + y (3) x 2 + y 2 = 1 2x 2 x + y 2 59

7 x > 3 x2 3x + 1 x 3 19 19 = k x (x > 0) x 2 +4 60

8 y = x 2 P (a, a 2 ) Q P Q R P P QR 61

9 1 cm ABCD AB, AC, AD P, Q, R 1 cm, 2 cm, 4 cm P, Q, R A t P, Q, R (1) AP = AQ = AR t (2) 1 4 < = t < = 1 2 P Q2 + QR 2 + RP 2 62

10 13x 2 8xy + 7y 2 = 1 x 2 + y 2 63

I II A B c 1 10 2 0.2 1 0. 5 7 64

3 (1) n 2n 3 3n 2 + n 6 (2) m, n m 3 n mn 3 6 (3) n n 4 + 2n 3 + 11n 2 + 10n 24 65

4 n 20 (1) n n 2 + n + 1 (2) n > 1 n 7 n 42 20 [ ] n n 9 n 3 9 66

5 (1) xy 3x 4y = 2 x, y (2) 5xy x 2y = 3 x, y 6 (1) x, y, z 1 x + 1 y + 1 z = 1, (x < = y < = z) (x, y, z) (2) x, y, z 1 x + 1 y + 1 z < 1, (x < = y < = z) 1 x + 1 y + 1 z 67

7 21 (1) x, y 1 < x < y (1 + 1 x )(1 + 1 y ) = 5 3 x, y (2) x, y, z 1 < x < y < z (1 + 1 x )(1 + 1 y )(1 + 1 z ) = 12 5 x, y, z 21 [ ] ABC tan A, tan B, tan C 68

8 (1) 6x 2 + xy 2y 2 5x + 6y + k x, y k (2) 6x 2 + xy 2y 2 5x + 6y 20 = 0 (x, y) 69

9 (1) 2 2223 (2) 3 + 2 2 22 [ ] 23 [ ] (1) 3 2 (2) 2 + 3 (3) log 2 3 (4) tan 1 70

10 x 0 < = y < 1 x y y < x > a {a n } a n 24 (i) a 1 =< a > (ii) { an 0 a n+1 =< 1 a n > a n = 0 a n+1 = 0 (1) a = 2 {a n } (2) n a n = a 1 3 a 24 [ ] 2 + 2 + 2 + 71

11 n n n 5 2017 2017 1 2 3 4 5 6 7 8 9 0 12 n n n 2 + 2 n = 3 25 26 25 n n + 2 11 13 26 [ ]n n, n + 2, n + 4 n = 3 72

13 a, b, c a 3 + 2b 3 + 4c 3 = 2abc (1) a, b, c (2) a = b = c = 0 27 27 73

14 8 n (1) 3 2n 1 (2) 3 2n 1 + 1 (3) 3 n 74

15 (2 4 )! 2 28 16 p n (p n )! p 28 [ ]219! 2 75

ax + by = c 17 5x 3y = 4 x, y 29 18 212x + 13y = 2 x, y 29 5x 3y = 1 x, y 5x 3y = 0 x, y m 76

19 (1) 14x 11y = 7 x, y (2) x, y x 2 + y 2 (3) 14x 12y = 7 x, y 20 2x + 3y 17 (x, y) 9x + 5y 17 (x, y) (x, y) 77

21 m n x = 3m + 5n x 22 xy x y (m, n) r 2 5 r 30 30 a, b am + bn m, n 78

23 2 f(x) = ax 2 + bx f(1), f(2) n f(n) 24 f(x) = 2x 3 + ax 2 + x a f(x) = 0 0 (1) a (2) x f(x) 6 79

25 a, b, c, d f(x) = ax 3 + bx 2 + cx + d f( 1), f(0), f(1) 3 f(x) = 0 80

26 f(x) = x 3 + ax 2 + bx + c (1) α f(x) = 0 α (2) k (> 1) k f(1), f(2),, f(k) k f(x) = 0 81

27 180 181 361 28 n, k k < = n 2k 2n 2k n k n k 82

29 p 31 (1) p C 1, pc 2, pc 3,, pc p 1 p (2) n n p n p 32 31 P x p 1 1 (mod p) x p x (mod p) 32 (m + 1) p = m p + p C 1 m p 1 + p C 2 m p 2 + + p C p 1 m + 1 n, p n 5 n (mod 5) n p n p+4 1 83

I II A B c tan θ = sin θ cos θ sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = 1 cos 2 θ 90 ± θ, 180 ± θ 90 sin cos 90 sin cos θ 10 tan tan θ = sin θ cos θ sin(90 + θ) = cos θ cos(90 + θ) = sin θ sin(180 + θ) = sin θ cos(180 + θ) = cos θ sin(90 θ) = cos θ cos(90 θ) = sin θ sin(180 θ) = sin θ cos(180 θ) = cos θ tan(90 + θ) = sin(90 + θ) cos(90 + θ) = cos θ sin θ = 1 tan θ a sin A = b sin B = c sin C = 2R R a 2 = b 2 + c 2 2bc cos A cos A = b2 + c 2 a 2 2bc S = 1 abc bc sin A = 2 4R r S = 1 r(a + b + c) 2 P (x 1, y 1 ) Q(x 2, y 2 ) S = 1 2 x 1y 2 x 2 y 1 a b S = 1 a 2 2 b 2 { a b} 2 s = a + b + c S = s(s a)(s b)(s c) s = a + b + c 2 2 84

sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β sin α sin β tan α ± tan β tan(α ± β) = 1 tan α tan β sin 2α = 2 sin α cos α cos 2α = cos 2 α sin 2 α = 2 cos 2 α 1 = 1 2 sin 2 α tan 2α = 2 tan α 1 tan 2 α sin 2 α 2 = 1 cos α 2 cos 2 α 2 = 1 + cos α 2 tan 2 α 2 = 1 cos α 1 + cos α sin 3α = 4 sin 3 α + 3 sin α cos 3α = 4 cos 3 α 3 cos α a sin θ + b cos θ = a 2 + b 2 sin(θ + α) a cos α = a2 + b 2 85

1 ABCD AB = 2, BC = 1+ 3, CD = 2, ABC = 60, BCD = 75 DA 2 a = 5, b = 6, c = 4 ABC 86

3 ABC 33 (1) a 2 tan C = c 2 tan A (2) b cos 2 A + c cos 2 C = b cos 2 B + c cos 2 A 33 87

4 ABCD AB = 2, BC = 3, CD = 4, DA = 5 (1) cos A (2) BD (3) ABCD 88

5 AB = 6 5 5, BC = 6 ABC A BC B D, E BD = 2, DE = 1, EC = 3 (1) AE (2) AD (3) AC 89

6 (1) ABC BC = a, CA = 3a 2, AB = 5a 4 a (2) ABC 3 3 (5a 4) a 90

7 cos 2 θ 4 cos θ = k k 0 < = θ < 2π 34 34 [ ] 0 < = θ < 2π a cos 3θ cos 2θ + 3 cos θ 1 = a θ a 91

8 f(θ) = 5 3 cos 2 θ + 3 sin 2 θ 4 sin θ cos θ (0 < = θ < 35 36 = π) 35 [ ] 5 cos θ + 6 sin θ (0 < = θ < π = 2 ) 36 [ ] y = a sin x + b cos x x = π 5 a, b 6 92

9 y A(0, 3) B(0, 1) x P (p, 0) AP B = θ θ p 37 37 [ ] y = 2x + 1, y = 1 x + 2 θ 3 93

10 0 < = α < 2π, 0 < = β < 2π sin α + cos β = 2, cos α + sin β = 2 38 (1) sin(α + β) (2) α, β 38 [ ] sin x + sin y = 1 2, cos x cos y = 1 3 cos(x + y) 94

11 tan α + tan β + tan γ = tan α tan β tan γ α + β + γ π 2 < α < π 2, π 2 < β < π 2, π 2 < γ < π 2 39 39 [ ]x + y = 3 π (tan x 1)(tan y 1) 4 95

12 (1) f(x) = 2 sin x (2) f(x) = sin(sin x) (3) f(x) = cos(sin x) (4) f(x) = sin(x 3 ) 96

13 O A(1, 0), B( 1 2, 3 2 ), C( 1 2, 3 2 ), P (cos θ, sin θ) ABP S 1 ACP S 2 S 1 S 2 P BC A 97

I II A B c a 1 n n a a n 1 a n a 0 1 x a x b = x a+b (x a ) b = x ab a m b a m b a x = y log a y = x a y a 0 < a < 1, 1 < a y 0 < y (1) log a 1 = 0 (2) log a a = 1 (3) log a M + log a N = log a MN (4) log a M log a N = log a M N (5) log a M N = N log a M (6) log M N = log a N log a M (7) log a b log b c = log a c (8) log a b = 1 log b a (9) a log a N = N 98

1 (1) 2 2 3 2 1 2 (2) ( 2 3 )2 ( 4 3 ) 3 (3) ( 4 5 ) 1 ( 5 2 )2 2 1 (4) a 5 (a 3 b 2 ) 1 (5) (xy 1 ) 2 (xy 1 ) 2 (6) 3 ab b a 3 b a 4 (7) log 2 3 log 2 9 8 + log 2 6 (8) log 2 6 log 6 12 log 12 16 (9) log 2 27 log 2 3 (10) 2 log 2 9 99

2 log 10 2 = 0.3010, log 10 3 = 0.4771 40 (1) log 10 1 (2) log 10 2 (3) log 10 3 (4) log 10 4 (5) log 10 5 (6) log 10 6 (7) log 10 7 (8) log 10 8 (9) log 10 9 (10) log 10 10 3 log 10 7 41 40 41 48 < 7 2 < 50 100

4 (1) log 2 3 (2) n log 2 n 101

5 (1) 2 3x+2 4 x + 2 x+1 5 = 0 (2) ( 4 + 15) x + ( 4 15) x = 8 102

6 x = 1 2 (5 1 n 5 1 n ) (x + x 2 + 1) n 7 5 x = 7 y = 10 10 x+y xy 42 42 [ ] a, b, c (a < b < c) x, y, z a, b, c a x = b y = c z = 30, 1 x + 1 y + 1 z = 1 103

8 3 2 3 log 9 8 43 9 x = 1 + log 1 2 3 2 x + 4 x 43 a log a x = x 104

10 2, 3 log 5 3, 4 log 5 2 11 1 < a < b < a 2 log a b, log b a, log a a b, log b b a, 1 2 105

12 (1) log 2 {log 3 (log 4 x)} = 1 (2) x 2 log 10 x = 1 100 x5 106

13 44 (1) 1 + log 9 (1 + x)(1 x) < = log 3 (3 + x) (2) log 2 (x 2 + 4) 2 log 4 x > = 4 44 [ ] > 0 107

14 (1) 2 10 = 1024 > 1000 log 10 2 > 0.3 (2) log 10 7 < 0.9 (3) log 10 13 > 1.1 108

15 45 (1) 18 35 (2) 18 35 16 ( 1 45 )54 0 99 45 [ ] 3 n n=0 109

17 f(x) = 3 2x a3 x+1 + 2a 2 18 a 4 x + b 2 x+1 a + 2 = 0 x = r, r log 2 3 a, b r 110

19 x > 0, y > 0, xy 3 log 10 x log 10 y 20 a 3 2 f(x) = 4x 2 x+1 a + a f(x) {x 1 < = x < = 1} f(x) 111

21 (x, y) 46 log x y < = log y x 46 [ ]x, y x 1, y 1 log x y + log y x > 2 + (log x 2)(log y 2) x, y (x, y) 112

I II A B c a n = a + (n 1)d S n = 1 2 n(a + l) = 1 2 {2a + (n 1)d} l = last 2a n+1 = a n + a n+2 a 1 = S 1 n > = 2 a n = S n S n 1 a n = ar n 1 S n = a(r 1) r 1 {a n+1 } 2 = a n a n+2 (1) (2) n n (a k + b k ) = a k + k=1 n n ca k = c k=1 k=1 n k=1 k=1 a k b k (1) (2) (3) (4) (5) n 1 = 1 + 1 + 1 + + 1 = n k=1 n k=1 n k=1 n k=1 n k=1 k = 1 + 2 + 3 + + n = 1 n(n + 1) 2 k 2 = 1 2 + 2 2 + 3 2 + + n 2 = 1 n(n + 1)(2n + 1) 6 k 3 = 1 3 + 2 3 + 3 3 + + n 3 = { 1 n(n + 1)}2 2 ar k 1 = a + ar + ar 2 + ar 3 + + ar n 1 = a(rn 1) r 1 n 1 a n = a 1 + k=1 b k 113

1 10 13 13 4 a n (1) a n (2) n S n (3) S n n 114

2 3 8 6 64 a n (1) a n (2) n S n (3) S n < = 10000 n 115

3 α, β, αβ (α < 0 < β) α, β 4 a 1, a 2,, a n (n > = 3) n a k = 55 k=1 116

5 x > = 0, y > = 0, y < = x + n n 6 x > = 0, y > = 0, y < = 1 3x + n n 47 47 [ ]x > = 0, y > = 0, y < = 2 x + 2n n 3 117

7 a n n S n S n = n 3 + 3n 2 + 2n (1) a n (2) n k=1 1 a k (3) n 2 k a k k=1 118

8 a n = 3n 1 (1) a 1 + a 2 + a 3 + + a n (2) a 2 + a 4 + a 6 + + a 2n (3) 2 a1 + 2 a2 + 2 a3 + + 2 an 119

9 n 54 2n 63 3n 10 n a n 10 3 < a n < 10 5 n < = n < = log 10 2 = 0.3010, log 10 3 = 0.4771 n = n = a a 3 2 120

11 (1) n (3k 2 + 4k + 5) k=1 (2) n (n + k) 2 k=1 121

12 1, 2, 3,, n i, j (1) i j (2) i 2 j 13 n, 2(n 1), 3(n 2),, (n 1) 2, n 1 n 122

14 n (1) 1 1 2, 1 2 3, 1 3 4, (2) 1 1 3, 1 3 5, 1 5 7, (3) 1 1 3, 1 2 4, 1 3 5, (4) 1 1 + 2, 1 2 + 3, 1 3 + 4, (5) 1 1 2 3, 1 2 3 4, 1 3 4 5, 123

15 48 (1) n n (1), (3, 5, 7), (9, 11, 13, 15, 17) (2) 2015 (3) n 48 [ ] 1 4, 7 10, 13, 16 (1) n n (2) n 124

16 n n 17 n n 125

I II A B c 1 (1) a 1 = 2, a n = 4a n 1 + 3 a n n (2) a 1 = 1, 2a n+1 + a n = 4 a n n (3) a 1 = 1, a n = a n 1 + 5 a n n (4) a 1 = 1, a n+1 = a n + n a n n 126

(5) a 1 = 1, a n+1 = a n + 2 n a n n (6) a 1 = 1 3, a n = a n 1 2a n 1 +1 a n n (7) a 1 = 3, a n = 2a n 1 + 2 n a n n (8) a 1 = 2, a n+1 = 3a n + 2 n a n n 127

(9) a 1 = 1, a n+1 = a n + 2 n a n n (10) a 1 = 3, na n = (n + 1)a n 1 2 a n n (11) a 1 = 1, a n = 2a n 1 n + 1 a n n (12) a 1 = 1, a n = 3 2a n 1 a n n 128

(13) a 1 = 1, a n+1 = 2a n 2 a n n (14) a 1 = 1, a 2 = 4, a n+2 5a n+1 + 6a n = 0 a n n (15) a 1 = 1, a 2 = 5, a n+2 6a n+1 + 9a n = 0 a n n (16) a 1 = 1, b 1 = 2, a n+1 = 2a n + b n, b n+1 = a n + 2b n a n n 129

2 {a n } {b n } n > = 3 a n = a n 1 + a n 2, b n = b n 1 b n 2 2 (1) {a n } {c n } {c n } (2) n > = 3 a n a 1, a 2, n (3) b 1 = 1, b 2 = 2 n > = 3 log 2 b n n 130

3 a n, b n (3 + 5) n = a n + b n 5 (1) a n+1, b n+1 a n, b n (2) c n = a n b n 5 cn (3) a n, b n 131

4 a n n S n 2a n S n = 3 n a n 5 a 1 = 1, a n = 2 n(n + 1) n a k (n > = 2) a n 49 k=1 49 [ ] {a n } a 1 n a n S n a 1 = 0, a 2 = 1, (n 1) 2 a n = S n (n > = 1) a n 132

n 6 a n a 1 = 1, a n+1 = ka k k=1 (1) n > = 2 a n+1 a n (2) a n n (3) n k=1 k a k+1 133

7 n 50 O C A B 50 [ ] ABC A, B, C P P A n P A a n a n [ ] A 1, A 2, A 3, A 4 X X A i n P i (n) (n = 0, 1, 2, ) P 1 (0) = 1 4, P 2(0) = 1 2, P 3(0) = 1 8, P 4(0) = 1 8 P 1(n) P 2 (n) 134

8 A, B, C, D Q Q A n Q n Q 2n = A a n D C A (1) a 1 B (2) a n 135

9 n ABCD EF GH P, Q P A Q C n P Q n + 1 P n Q n n P Q n + 1 P Q n (1) P Q P Q (2) n P Q r n (3) n P Q ABCD EF GH p n n P Q ABCD EF GH q n p n+1 p n q n q n (4) lim n p n D C A B H G E F 136

10 a n (n + 3)a n+1 (2n + 4)a n + (n + 1)a n 1 = 0 (n > = 2) (1) b n = a n+1 a n b n b n 1 (2) b n n b 1 (3) a 1 = 1 3, a 2 = 1 2 a n (4) (3) a n lim n (a n) n 137

I II A B c 1 51 1 2 + 2 2 + 3 2 + + n 2 = 1 n(n + 1)(2n + 1) 6 2 52 1 2 + 2 2 + 3 2 + + n 2 = 1 n(n + 1)(2n + 1) 6 51 n 3 (n 1) 3 (n + 1)n(n 1) n(n 1)(n 2) = 3n(n 1) 52 138

3 k n 1 k + 2 k + 3 k + + n k = n5 5 + n4 2 + n3 3 n 30 k 4 (cos θ + i sin θ) n = cos nθ + i sin nθ 139

5 19 n + ( 1) n 1 2 4n 3 53 53 (1) n 3 n+2 + 4 2n+1 13 (2) n 2 2n+1 + 3 ( 1) n 5 140

6 n > = 2 n 1 + 1 2 2 + 1 3 2 + + 1 n 2 > = 13 8 1 n + 1 2n 2 141

7 t = x + 1 x xn + 1 x n t n 142

(a + b) n = n C 0 a n + n C 1 a n 1 b + n C 2 a n 2 b 2 + + n C r a n r b r +... + n C n b n 54 8 (1) (2x 3y) 5, x 3 y 2 (2) (x + 2 x )7, x (3) (x y + 2z) 6, xy 3 z 2 (4) (x 2 2 x + 1)6, 54 n! p!q!r! ap b q c r p + q + r = n 143

9 k (1 + x + kx 2 ) 6 x x 3 x 4 x 4 k 10 (x + 5) 80 x 55 55 [ ](x + 3) 50 x 144

11 100 2000, 12345 0, 45 m 5m 4 12 (1) n C 0 + n C 1 + + n C k + + n C n = 2 n (2) n C 0 + 2 nc 1 + + 2 k nc k + + 2 n nc n = 3 n (3) n C 1 + 2 nc 2 + 3 nc 3 + + n nc n = 2 n 1 n 145

I II A B c 1 a = 1, b = 1 ( a + b) (2 a 3 b) 3 a + 2 b 2 A(8, 2), B(2, 8) y = kx P (x, y) AP, BP AP BP = 0 k 56 56 [ ] V V V 146

3 ABC 2 PA + 3 PB + 4 PC = 0 P P BC : P CA : P AB 57 A B C 57 3 b+4 c = 7 9 9 3 b+4 c 7 147

4 a ABC P 5 PA + a PB + PC = 0 AP = a+ AB + a+ AC AP BC D BC 1 : 8 a = AP = AD P AD : AB = 2 2, BC = 10, AC = 6 AB AC = AP 2 = 148

5 ABC AB M AC 2 : 3 N B N C M P A M N P B (1) AP AB AC C (2) AP BC Q AQ (3) 149

6 OA = 13, OB = 5, AB = 4 OAB OA, OB D, E AE BD F AE, BD 7 : 1, 5 : 3 (1) OA OB (2) OF OA, OB (3) O F C (4) C P AP BP 150

7 AB = 4, BC = 6, CA = 5 ABC I AB = b, A AC = c B (1) AI b, c C (2) AI = k( b + c b c ) k (3) IBC 151

8 ABC O R = 1 4 OA + 5 OB + 6 OC = 0 AB 9 ABC O OA + OB + OC = 0 ABC 152

10 ABC H AB = 4, AC = 5, BAC = 60 AH BC Q AB = b, AC = c (1) AQ b, c (2) AH b, c 153

11 60 OA OB OA = 3, OB = 2 P P A = P B OP 58 AOB 58 [ ] OAB a = OA, b = OB a = 3, b = 5, cos ( AOB) = 3 5 AOB B 10 O a, b 154

12 ABC O G H OG : GH = 1 : 2 155

13 ABCD EF GH F G M AB = b, D AD = d, AE = e A H C E B G (1) AM F (2) AM BDE N AN 156

14 OABC OA 1 : 2 P BC Q P Q 1 : 3 R O C A (1) OR B (2) OR ABC S OS (3) CR OAB T OT 157

15 OABC OA, OB, AC, BC K, L, M, N O C A (1) K, L, M, N α B (2) OC P AB Q P Q R α 158

16 ABCD AB 1 : 2 P P C 1 : 2 Q AQ BC R BD S BCD G AG QS T AT AB = b, AC = c, AD = d 159

17 OABC AOB = AOC = π 2, BOC = π, OA = OB = 2, OC = 1 3 A, B, C P OP = p OA = a, b, OC = c p s, t OB = p = (1 s t) a + s b + t c (1) p a, p b, p c s, t (2) P AOP = BOP = COP s, t 160

(3) (2) P AP BC Q BP AC R BQ : QC AR : RC (4) (2) P OABP, OBCP, OCAP 59 59 [ ] AOB = AOC = 60, BOC = 90, OB = OC = 1, OA = 2 OABC O ABC H OH 161

18 O PQ = PR = 4 QR = 3 P, Q, R QR M (1) OM (2) OP OM (3) MP OM (4) P M l O l H OH OP OM 162

19 O ABCD EF GH AB = b, AD = d, AE = e (1) CF 2 : 1 I AI (2) AI J AJ 163

20 OABC (1) OA BC, OB AC, OC AB (2) 164

I II A B c 1 A(1, 1, 1) xy P, Q P Q (1) P AQ (2) P AQ 165

2 l : x 3 = y 4 = z m : x 2 = y + 1 2, m T ST S, z = 0 l S T 166

3 A(1, 0, 0) B(0, 2, 0) C(0, 0, 3) D(4, 4, 4) (1) A, B, C α (2) α D (3) ABC (4) ABCD 167

4 S xy (x 1) 2 + (y 2) 2 = 16 S ( 2, 6, 2) S 168

5 A(2, 1, 0), B(1, 0, 1), C(0, 1, 2), D(1, 3, 7) A, B, C D E E 169

6 α : x + 2y + z = 5 A(0, 0, 2), B(2, 1, 0) (1) α A (2) AB α (3) P α AP + BP P 170

7 A(0, 0, 1), B(2, 1, 0), C(0, 2, 1), D(0, 2, 1) (1) C AB CH H (2) P xy Q AB DP, P Q DP +P Q P, Q 171

8 A( 2, 0, 0), B(0, 2, 0), C(0, 0, 2), D(2, 1, 0) A, B, C T (1) D T H (2) T A, B, C S (3) P S DP P 172

9 xyz x + y = 4, z = 1 α x 2 + y 2 + z 2 = 4 α 10 x = y = z x + y + 2z = 0 π 3 60 60 [ ] A(0, 2, 0), B(1, 3, 2), C(3, 3, 4) A AB AC l AP B = 30 l P 173

11 (0, 1, 3) x 2 + y 2 + (z 1) 2 = 1 61 (1) (2) xy xy 61 [ ]xyz A(5, 0, 0), B(4, 1, 0), C(5, 0, 2) T T A 2 U (1) P U P AB = θ (0 < = θ < 2π) P (u, v, r) θ (2) D(10, 0, 0) P yz Q(0, Y, Z) Y Z θ (3) Q 174

I II A B c f f(b) f(a) f(a + h) f(a) (a) = lim = lim b a b a h 0 h {f(x)g(x)} = f(x) g(x) + f(x)g (x) {f(x)g(x)h(x)} = f(x) g(x)h(x) + f(x)g (x)h(x) + f(x)g(x)h (x) { f(x) g(x) } = f (x)g(x) f(x)g (x) {g(x)} 2 {f(x) n } = nf(x) n 1 f (x) 1 (1) y = (x 3 + 2x 2 ) 5 (2) y = (x + 3) 3 (x 2 + 1) 2 175

2 x 3 + ax + b lim x 1 (x + 1) 2 a, b 3 f(x) f(x) = 1 5 x5 1 3 x3 + x 1 f(3 + 3h) f(3 2h) lim h 0 h 176

4 f(x) = x 3 + 3ax 2 + 3bx + c x = α x = β (1) α + β, αβ a, b (2) f(α) + f(β) a, b, c (3) y = f(x) A, B AB 177

5 P (x) (x 1) 2 4x 5 x + 2 4 P (x) (x 1) 2 (x + 2) 62 6 63 62 [ ]x n + ax + b (x 1) 2 a, b 63 [ ] 178

7 f (x){f (x) 3 2x} = f(x) + 3(x + 1) f(x) 64 8 f(x) = x 4 + 4px 2 p 2 x + q x = 1 p, q 64 [ ] f (x){f (x) 3x 2 x 2 3 } = 9f(x) 2x2 20 3 x 19 n f(x) 3 f(x) 179

9 x + y + z = 0, x 2 + y 2 + z 2 = 6 x 3 + y 3 + z 3 180

I II A B c 1 y = x 2 + 3x + 2 (0, 2) 2 y = (x 1) 2 y = (x + 1) 2 65 65 [ ]a C 1 : y = x 2, C 2 : y = x 2 4ax + 4a (1) C 1 C 2 l (2) C 1, C 2 l 181

3 y = x 4 3x 2 + 2x + 1 182

4 (1, 2) y = x 3 3x + 3a 2a 2 a 5 x 3 + x 2 x + a = 0 66 66 [ ] (1) x 4 4x 3 2x 2 + 12x a = 0 (2) y = x 3 9 2 x2 + 5x y = x + a (3) x 3 + 3px p = 0 183

6 y = x 3 x ( 1, a) a 67 7 a y = x 3 x y = x 2 + a 67 [ ] y = x 2 (a, 2) a 184

8 k x 3 13x + k = 0 k 68 9 xy y = x 2 + 4 x 68 [ ]k x 3 5x 2 + 2x k = 0 k 185

10 n f(x) = 2x n+1 4x n + 3 69 (1) f( 3 2 ) (2) f(x) = 0 69 [ ] (1) n 2 n > = 2(n + 1) (2) n x n nx + 1 = 0 2 186

11 y = x 4 2x 2 + 1 187

I II A B c 1 1 < = x < = 1 f(x) f( 1) = f(0) = 1, f(1) = 2 y 1 1 0 1 x 1 2 1 1 f(x)dx > = 1 188

2 1 (1) f(x) = ax 2 +bx+c (a > 0) g(x) f(x)g(x)dx = 1 0 a, b, c (2) f(x) 1 1 {f(x)} 2 dx = 2 5 a, b, c 189

3 70 (1) y = x 2 y = x + 1 (2) y = 2x 2 y = x + 6 4 xy x < = 2, y > = x, y < = 3 4 x2 3 2 β 70 1 6 = k(x α)(x β)dx = α k (β α)3 6 190

5 y = x 3 2x (0, 2) 6 a 2 < a < 4 4 0 x a dx = 17 4 a 71 1 71 [ ] t F (t) = x 2 t 2 dx 0 (1) F (t) t (2) t > = 0 F (t) t 191

1 7 S = x 2 + 2ax dx a > = 0 S 2 a 72 72 [ ] a 0 < a < 1 x f(x) = 4x 3 3x + a (1) f(x) = 0 0 < x < 1 β (2) f(x) = 0 0 < x < 1 α, β (α < β) f (x) dx = a α a 192

8 y = x 2 (1, 2) S(k) 9 y = x 2 + 4 y = x 2 193

10 y = mx + k y = x 2 36 73 73 [ ] y = x 2 P, Q P Q P Q R 194

11 y = x 2 A(a, a 2 ) B(b, b 2 ) (a < b) l A, l B C AB M y = x 2 AB l D D y = x 2 AB S 1 y = x 2 l A, l B S 2 74 M B A S 1 D S 2 C (1) C, D, M a, b (2) MD : DC (3) S 1 : S 2 74 [ ] C : y = x 2 A 1 (a 1, a 2 1 ), A 2(a 2, a 2 2 ), A 3(a 3, a 2 3 ), A k+2 (k > = 1) C A k A k+1 a 1 < a 2 A k A k+1 A K+2 T k A 1 A 2 C S (1) T k+1 T k (2) lim n n T k S k=1 195

12 C : y = x 2 P (a, a 2 ) l 1 P Q l 2 l 1 a > 0 (1) l 1 (2) l 2 (3) P Q C S(a) (4) S(a) a (5) l 1, l 2 C T (a) 196

13 C : y = x 2 + ax + b P 1, P 2 C l 1, l 2 l 1 l 2 C, l 1, l 2 a, b 197

I II A B c 1 (1) y = ax 3 + bx 2 + cx + d (2) y = ax 3 + bx 2 + cx + d A, B y = Ax 3 + Bx 198

(3) y = Ax 3 + Bx (4) y = Ax 3 + Bx P l y = Ax 3 + Bx P Q P x α Q x β α : β 199

2 C 1 : y = x 3 x A(a, a 3 a) C 1 l a 0 (1) C 1 l A x (2) C 1 l S 1 (3) l C 2 : y = x 3 x + c (c 0) C 2 B B x c (4) C 2 l B C x (5) C 2 l S 2 200

3 C : y = x 3 x P l P x a (1) l C Q y R P R L 1 RQ L 2 L 1 : L 2 (2) C l y S 1 y S 2 S 1 : S 2 201

4 xy C : y = x 3 + ax 2 + bx + c P l P Q C l C Q m C 75 75 [ ]f(x) = x 4 + ax 3 + bx 2 + cx + d y = f(x) y (1) a, b, c, d (2) f(x) 202

5 f(x) x, y f(x + y) = f(x) + f(y) + xy(x + y) x = 0 f(x) a (1) f(0) (2) f (x) (3) f(x) 203

6 f n (x) = x n+1 (n + 1)x + n (n = 1, 2, 3, ) (1) f n (x) n (x 1) 2 (2) y = f 1 (x), y = f 2 (x) (3) y = f 2n 1 (x) y = f 2n (x) S n n 204

7 f 1 (x), f 2 (x), f 3 (x), f 1 (x) = x 2 + x 2, x 2 f n+1 (x) = 2x 3 + x 4 + x 0 tf n (t)dt (n = 1, 2, 3, ) (1) f n (x) x (2) f n (x) 205

8 O(0, 0), A(1, 0), B(1, 1), C(0, 1) y = 1 + 2kx 3k 2 x 2 k 206

9 P 1 (x) C 1 1 P 1 (x)cdx = 0 P 2 (x) Q(x) 1 1 P 2 (x)q(x)dx = 0 P 3 (x) R(x) 1 1 P 3 (x)r(x)dx = 0 P 1 (x), P 2 (x), P 3 (x) P 1 (x), P 2 (x), P 3 (x) 207

10 f(x)dx... f(x)dx = A (1) f(x) = x + 1 + 2 g(t)dt, g(x) = 2x 3 + 1 0 0 f(t)dt f(x), g(x) (2) f(x) = 3x + 1 0 (x + t)f(t)dt f(x) 208

11 f(t)dt f(x) = d dx x a f(t)dt a a f(t)dt = 0 (1) f(0) = 1, x 0 f(t)dt = xf(x) + x 2 + x 3 f(x) (2) f(x) = 1 + x 0 1 g(t)dt, g(x) = x(x 1) + f(t)dt f(x), 1 g(x) 209

(3) x f(x), g(x) x f(t)dt = xg(x) + ax + 2 a 1 g(x) = x 2 2x 1 0 f(t)dt + 1 a f(x), g(x) 210

12 f(x)dx (1) f(x) = x+1 x (t 3 3t 2 + 2t)dt (2) f(x) = x+a x (t 4 t 2 1)dt a i. a = 2 f(x) ii. f(x) x a 211

13 f(x) = x 3 + x 2 + 1 1 (x t) 2 f(t)dt f(x) 76 76 [ ] f(x) C x 1 f(y)dy + (x + y) 2 f(y)dy = x 2 + C 0 0 f(x) C 212

I II A B c (x 0, y 0 ) ax + by + c = 0 h = ax 0 + by 0 + c a2 + b 2 y (x 0, y 0 ) l ax + by + c = 0 O x (x p) 2 +(y q) 2 = r 2 (s, t) (s p)(x p)+(t q)(y q) = r 2 1 (1) x 2 + y 2 = 25 ( 3, 4) (2) (x 1) 2 + (y 2) 2 = 25 (4, 6) (3) x 2 + y 2 4x 2y 5 = 0 (5, 2) 213

(x p) 2 +(y q) 2 = r 2 (s, t) (s p)(x p)+(t q)(y q) = r 2 (s, t) 2 (1) x 2 + y 2 = 4 (4, 2) (2) x 2 + y 2 = 5 (1, 3) 214

S 1 = 0 S 2 = 0 a S 1 + b S 2 = 0 S 1 + k S 2 = 0 S 2 = 0 3 x 2 + y 2 2x = 0 x 2 + y 2 4y 4 = 0 (1) (2) (3) (1, 0) 215

4 x 2 + y 2 + 2x + 4y 4 = 0 7x y + 2 = 0 ( 1, 2) 5 x 2 + y 2 = 2, (x 1) 2 + (y + 1) 2 = 1 y = x 77 77 [ ]xy A( 1, 4), B(2, 5) y = 1 2 x (1) P (2) r 216

6 y = x + 1 x 2 + y 2 = 4 78 7 y = 2x + 1 x 2 + y 2 2(x + y) = 0 78 = = 217

8 ( 3, 2), (0, 1), (3, 0) 79 9 A(0, 0), B(14, 0), C(9, 12) 79 AP BC 218

10 (x 3) 2 + (y 2) 2 = 26 y = 3 2 x + k 52 k 11 x 2 + y 2 = 1 16, (x 1)2 + y 2 = 1 4 219

12 y = x 2 + (2a 1)x a 2 + 1, y = x 2 (a 2)x + a 2 + a 13 P (a, b) x y l l x y S l 220

14 y = x 2 2x + 3 (1) x = 2 (2) ( 1, 1) (3) y = 2x + 1 221

15 l : ax + (1 a)y 1 = 0 C : x 2 + y 2 4x + 2y 4 = 0 (1) l C (2) C l (3) l C P, Q P Q a 222

16 O xy C : (x 2) 2 + y 2 = 1 l : y = kx C l y A, B A, B m C l P, Q l m R (1) C l k (2) OP OQ (3) OR(OP + OQ) 223

17 C 1 : x 2 + y 2 2ax 2ay + 8a 16 = 0, C 2 : x 2 + y 2 = 4 (1) C 1 a (2) C 1 C 2 224

I II A B c 1 (4, 2) y = x 2 1 : 2 P 2 3x 2y + 2 = 0 2x + 3y 4 = 0 225

O r O P OP OP OP = r 2 P O O y P O P x P, P (a, b), (A, B) a = A A 2 + B 2, b = B A 2 + B 2 a 2 + b 2 = 1 A 2 + B 2 226

3 P, P (a, b), (A, B) a = A A 2 + B 2, b = B A 2 + B 2 (1) O OP, OP (2) O, P, P (3) P 2x + y = 4 P 227

4 x, y, X, Y X = x x 2 + y 2, Y = y x 2 + y 2 P (x, y) (4x + 3y 5)(4x 3y + 5) > 0 Q(X, Y ) 80 80 [ ]xy r (r > 0) (a), (b) (a) OP OQ (b) OP OQ = 1 (1) OA (2) (1) l l r 228

5 xy l : x + ty = 3t, m : tx y = 3 (1) t l m t (2) t l m 229

6 (1) xy(x y)(x 2 + y 2 4) > 0 (2) ( x + y 1)(x 2 + y 2 4) < = 0 7 a a > = 0 ax + (2 a)y = 2 230

8 l : y = ax + a 2 5 C : y = x 2 (1) a l (2) l C a a l 231

9 xy 1 < = y < = 1 D P O C C D P 10 x 2 + y 2 = 4 y = 1 (1) (2) ( 3, 1), ( 3, 1) 232

11 A(1, 0), B( 1, 0), C(0, 1) AP C = BP C P P A, B, C 12 xy y = x 2 {(x, y) : a < = x < = a + 1, b < = y < = b + 1} (a, b) 233

I II A B c 1 180 A B 17 B C 22 C A 20 A B 98 B C 107 C A 116 A B C 42 A, B, C A B C 2 a 1 < a 2 < a 3 < a 4 < a 5 a k A = {a 1, a 2, a 3, a 4, a 5 } a 2 k B = {a2 1, a 2 2, a 2 3, a 2 4, a 2 5} A B = {a 1, a 4 }, a 1 +a 4 = 5 A B 77 a 1, a 2, a 3, a 4, a 5 234

P Q Q P P Q Q P P P < > = = 3 a, b (1) a 2 + b 2 a, b (2) a 2 + b 2 a, b 235

a b (1) a > b A B > A B (2) a < b B A A < B (3) a = b A = B (4) a b A B 236

4 (1) x = 2 x 2 5x + 6 = 0 (2) x 2 3x + 2 = 0 x 2 5x + 6 = 0 (3) x 3 6x 2 + 11x 6 = 0 x 2 5x + 6 = 0 (4) xy = 0 x = y = 0 (5) x 2 + y 2 = 0 x = y = 0 (6) (x y)(y z) = 0 x = y = z (7) x 2 > y 2 x > y (8) a, b ab (9) x > 0 y > 0 xy > 0 (10) a + b < a + b ab > 0 (11) x > 0 y > 0 xy > 0 (12) 0 < = x < = 1 0 < = y < = 1 0 < = x 0 < = y xy < = 1 (13) 0 < = x < = 1 0 < = y < = 1 0 < = x 0 < = y xy < = 1 (14) x + y < 1 x + y < 1 (15) x 2 + y 2 < 1 x + y < 1 (16) x + y < 1 x y < 1 x + y < 1 (17) (18) A = 60 (19) (20) A, B A B = A A B = B (21) n n 2 12 n 12 (22) T T (23) a, b a + b = a + b ab > = 0 (24) a, b, c a + b + c = a + b + c ab + bc + ca > = 0 (25) (26) 237

5 m, n x A = x 3 + mx 2 + nx + 2m + n + 1 (1) x B B = x 2 2x 1 A B Q R Q = x + (m + ), R = (2m + n + )x + (3m + n + ) x = 1 + 2 B A 1 m, n m =, n = (2) x A i. m ii. n iii. m n iv. m v. n vi. m n 238

6 a, b x A = x 4 + (a 2 a 1)x 2 + ( a 2 + b)x + b 3, B = x 2 x a A B Q R Q = x 2 + x + a, R = (a + b)x + a + b (1) R = x + 7 a = a = (2) a < 1 2 x Q > 0 (3) a + b = 0 A B 239

7 a, b ( a + b + a b ) 2 = 2(a 2 + b 2 + ) ( a + b + a b ) 2 = 4a 2 ( a + b + a b ) 2 = 1 2 ( a + b + a b ) = b (1) a 2 (2) b 2 (3) 4a 2 (4) 4b 2 (5) ab (6) ab (7) 2ab (8) 2 ab (9) a 2 b 2 (10) b 2 a 2 (11) a 2 b 2 (12) a 2 < = b 2 (13) a 2 > = b 2 (14) a < = b (15) a < = b (16) a > = b (17) a > = b 240

8 n p, q, r, s p : n q : n 10 r : n s : n r r s s p r q r s p s q s U p P r R s S P, R, S (1) (2) (3) (4) U U U U P R P R P R P R S S S S 241

9 a, b p, q p : (a + b) 2 + (a 2b) 2 < 5 q : a + b < 1 a 2b < 2 (1) q = p (1)a = 0, b = 0 (2)a = 1, b = 0 (3)a = 0, b = 1 (4)a = 1, b = 1 (2) p = q (3) p q 242

10 100 A, B, C A 69 B 46 B, C 21 A, C 88 B, C 50 A, B, C 61 A B C (1) A (2) B (3) C (4) A, B, C (5) A, B, C 243

I II A B c 1 D C A B H G E F (1) (2) (3) 244

2 (1) (2) (3) (4) (5) 245

3 12 81 (1) (2) (3) (4) (5) 81 [ ] 246

4 5 (1) A B (2) A B (3) 5 5 (1) A B C (2) A B C (3) 247

6 A B 82 D B C A (1) (2) C (3) C D (4) C D 82 [ ] 1 m A B B C A (1) (2) C (3) 5 m (4) 4 m 248

7 A B B A 8 A B B A 249

9 (1) (2) 10 x + y + z = 10 x, y, z 83 (1) x, y, z (2) x, y, z 83 [ ]x + y + 2z = n (n ) x, y, z 250

11 84 (1) (2) 84 [ ] 1, 2, 3,, 100 251

12 13 14 n 252

15 n (1) (2) (3) 253

16 n n n k k m (1) m p(m) m, n, k (2) m q(m) m, n, k (3) k n 254

17 n X 85 (1) X P n (2) X Q n (3) X R n 85 [ ] n X n a n = X 1 X 2 X n n a n < = 9 255

18 (1) (2) n 256

19 ABCD BD 1 2 A C C D B A 20 ABCD 1 2 A B 86 A D B C 86 ABCD p A B 257

21 15 n P n (1) P n (2) P n n P n 22 n P n (1) P n+1 P n (2) P n 258

23 n n n > = 3 87 24 n 1 n n > = 2 87 n 259

25 A n A n B B n C C B A 260

I II A B c 1 2 3 ABC G G y BC y A, B y = kx 2 (k > 0) k G y C A G B O x 261

2 a > 0 xy x l : y = a U : y = x 2 (1) (0, s) r U s, r (2) l U D D y r 262

3 y = x 2 +2 x 263

4 y > = x 2 D D y C 1 n C n C n C n D y C n+1 C n a n b n = a 1 + a 2 + + a n C 3 C 2 C 1 (1) a 1 (2) n > = 2 a n b n 1 (3) a n n 264

5 xy y > = x 2 D D l 45 y t 88 y O x 88 [ ]xy y > = x 2 D D t x y t 265

6 a > 1 a (a, a) A y = 1 x (x > 0) 1 P (t, t ) t > 0 f(t) AP f(t) = AP 2 (1) f(t) t a (2) f (t) = 0 t (t > 0) (3) AP P AP 266

7 xy y = x 2 P, Q, R P QR a P, Q 2 a 89 89 [ ] y = x 2 P, Q, R P QR (1) P, Q, R x p, q, r p 2 + q 2 + r 2 pq + qr + rp (2) P QR 267

I II A B c 1 A, B, C, D, E, F, G, H AB = a, AD = b, AE = c H G E D F C A B (1) AG EBD M AM : MG (2) EBGD 268

2 ABCD A, B, C x, y, z AB = 2l 1, BC = 2l, CA = 2l + 1 (l > 2) ABCD V (l) 90 lim l 2 V (l) l 2 90 A(a, 0, 0), B(0, b, 0), C(0, 0, c), D(a, b, c) 269

3 BC = DA = a, CA = BD = b, AB = CD = c ABCD AB, BC, CD, DA, CA, BD P, Q, R, S, T, U (1) P R, QS P R QS (2) P, Q, R, S, T, U a, b, c 270

4 ABC ABC 271

5 V α S S 272

I II A B c 1 a sin A = A b sin B = c sin C = 2R O R C B 2 a 2 = b 2 + c 2 2bc cos A 273

3 ax + by + c = 0 (x 0, y 0 ) ax 0 + by 0 + c a2 + b 2 4 r (r > 0) x 2 + y 2 = r 2 (a, b) ax + by = r 2 274

5 α, β sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β sin α sin β 91 E D α O β α A B C 6 (cos θ + i sin θ) n = cos nθ + i sin nθ 91 e iθ = cos θ + i sin θ 275

7 log a b + log a c = log a bc (a > 0, b > 0, c > 0, a 1) 8 log a b = log c b log c a (a > 0, b > 0, c > 0, a 1, c 1) 276

9 n k=1 k 2 = 1 n(n + 1)(2n + 1) 6 277

10 ABC S S = 1 OA 2 2 OB 2 { OA OB} 2 11 OA = (a 1, a 2 ), OB = (b 1, b 2 ) θ OA OB = OA OB cos θ OA OB = a1 b 1 + a 2 b 2 278

12 α, β (α < β) β α (β α)3 (x α)(x β)dx = 6 279

13 (a + b) n = n C 0 a n + n C 1 a n 1 b + n C 2 a n 2 b 2 + + n C r a n r b r + + n C n b n 280

I II A B c a 2a (1) (2) O (3) (4) 281

a + b = 180 a b a A D A D B C B C BAC = BDC BAD + BCD = 180 ABCD AP B = T AB P B S A T 282

P A P B = P C P D B A D C P P A P B = P C 2 B A C P A P B = P C P D P D A C P B 283

1 DAE = 56 DOE A D O E B O C 2 ABC A BC ABC D, E AB = 3, AC = 5, BC = 7, BAC = 120 AD A B D C E 284

3 ABC I O r R E A F O I B C (1) BD : F I = DE : IA D (2) ID = BD (3) AI ID r, R 285

4 ABCD AB = 10, AD = 6, BAC = DAC AC BD E AE = 15 4 B 10 15 4 A E 6 D C (1) AC (2) BE : ED (3) BD 286

5 A BC D AD BAC C D B A 6 ABC A BDA = BAC = CEA D, E CE F D B A E F C (1) ABF (2) DA = AE 287

7 ABC 10 AD = 5, AE = 3 ADP E Q AP ABC 92 A D E P B C Q (1) AP (2) P Q 92 BQ + CQ = AQ 288

8 ABCD AB = 2, BC = 5 + 1, CA = 2 2 ABD : BCD 5 1 : 1 (1) ABC (2) AD : CD (3) CD 289

9 ABC AB = 7, BC = 3 ABC I AI BC D BI AC E C, E, I, D (1) BCA = AI = B + A BCA = CA = BD =, BI BE = (2) ABC ABC ABC 290

10 ABCD AB = a, BC = b, CD = c, DA = d ABC = α, CDA = β S = 1 2 (a + b + c + d) 93 a A d β D c B α b C (1) AC a, b, α AC 2 = (2) cos β = cos α cos α a, b, c, d cos α = 2(ab+cd) (3) sin 2 α = (1 cos α)(1 + cos α) sin α > 0 sin α = (S a)(s b)(s c)(s d) (4) T sin β = sin α T = ABC + ADC = 93 [ ] a, b, c ABC S s = a + b + c 2 S = s(s a)(s b)(s c) 291

I II A B c I II A B 1 (1) (x + 2y 2)(x + 2y + 1) (2) (2x + 3y 1)(4x + 2y + 1) (3) (x + y + z)(xy + yz + zx) (4) (x + y + 1)(x 2 + y 2 + 1 xy x y) (5) (x + 1)(x 1)(x 2 + x + 1)(x 2 x + 1) (6) (x + a)(x 3)(x + 1) (7) 3(a b)(b c)(c a) (x 1)(x + a)(x + a + 3) 2 (1) 2+ 2 6 4 (2) 2 3+3 2 30 12 (3) 3 + 2 (4) 2 3 (5) 3 6 (6) 10+ 6 2 (7) 3 2 10 2 (8) 2 3 + 1 (9) 6+ 2 4 3 7a 2 a + 1 a 4 k = 0, 1 0 k = 2 1 k = 3 a + b + c 5 (1) 0 (2) 27 6 6 + 16 3i 7 (x 1)(y 1)(z 1) = (xyz 1) + {(x + y + z) (xy + yz + zx)} = 0 8 8 A = 1, B = 7, C = 6, D = 1 9 6 292

2 1 (1) 13 (2) 6 2 (1) (x + 1)(x + 2)(x + 3) (2) (3x + 2)(x 2 + x + 1) 3 a = 2, b = 14 4 2x 5 7x 12 6 a = 2, b = 3, c = 3 7 (1) x 1 (2) x 1 (3) 5x 5 8 x 2 + 4x 4 9 10 3x + 3 11 a = 9, b = 8, c = 9, d = 1 0 12 ±1, ± 1 2, ± 1 4 13 x = ±1 14 (±1, 0, ±1), (±2, 2, ±2), (±3, 4, ±3) (1) α 3 = 2 α (2) α 3 + α 2 = (α 1)(α 2 + α + 2) α = 1 1 (1) a = 0 a = 1 a 0, 1 1 a (2) (x, y) = (2, 1), ( 1, 1), ( 1, 2) (3) (x, y, z) = ( 1, 1, 2), ( 1, 2, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (2, 1, 1) 293

(4) (x, y, z) = ( 1, 1, 2), ( 1, 2, 1), (1, 1, 2), 2 k = 4, x = 3 3 (1) a = 10 (1, 2, 1), (2, 1, 1), (2, 1, 1) (2) a = 8 4 k = 3, 2 5 27 2 995 6 k < = 1, 3 < = k 7 a < 4 x = 4 a+ 4 a 2 1± 8 (a, b) = (1, 0) x = 0, 1, 3i 2 (a, b) = ( 1 2, 3 2 ) x = 1± 15i 1± 4, 5i 2 1 (1) 3 29 2 < = x < 1 5 1+ = 2, 5 < 2 = x < 3+ 29 = 2 (2) 1 < = x < = 2, 3 < x (3) 5 41 2 < = x < = 1+ 41 2 (4) x < = 7, 2 3 < = x < = 7, 2 + 3 < = x (5) a < 1 a < x < 1 a = 1 a > 1 1 < x < a 1 (6) 0 < a < 1 x < a, a < x a = 1 x = 1 1 < a x < 1 a, a < x 2 (1) (x y 2 )2 + 3 4 y2 > = 0 1 2 {(x y)2 + x 2 + y 2 } > = 0 x = y = 0 (2) (x y+z 2 )2 + 3 4 (y z)2 > = 0 1 2 {(x y)2 + (y z) 2 + (z x) 2 } > = 0 x = y = z 3 x = 3, y = 2 3 4 x = 1 14, y = 1 7, z = 3 14 1 14 5 ( x y) 2 > = 0 6 3 x = X, 3 y = Y, 3 z = Z X 3 + Y 3 + Z 3 3XY Z = (X + Y + Z)(X 2 + Y 2 + Z 2 XY Y Z ZX) = 1 2 (X + Y + Z){(X Y )2 + (Y Z) 2 + (Z X) 2 } > = 0 (1) (1, 2, 3) 294

(2) x 3 + y 3 + z 3 > = 3 3 x 3 y 3 z 3 = 3xyz x, y, z 3xyz > xyz n = 3 (x, y, z) 7 (e)48, (f) 8 0 4 24 3 2 2 9 4 10 a + b + c > = 3 3 abc, ab + bc + ca > = 3 3 a 2 b 2 c 2 11 (1) (a 1)2 (a 2 +a+1) a 2 > = 0 (2) (a 1)2 (a 4 +a 3 +a 2 +a+1) a 3 > = 0 12 (1) (a 1)(b 1) > 0 (2) abc + 2 > ab + c + 1 = a + 1 + c > a + b + c 13 2(a 2 + b 2 + c 2 ) (c a) 2 = (a + c) 2 + 2b 2 > = 0 a + c = 0, b = 0 a 2 + b 2 + c 2 = 2(ab + bc + ca) 3(a 2 + b 2 + c 2 ) 2(c a) 2 = 2(b a)(c b) < 0 14 (1) ( a+b 2 )2 < a2 +b 2 2 (2) ( a+2b 3 ) 2 < a2 +2b 2 3 (3) sa 2 + tb 2 > (sa + tb) 2 (4) log( a+b log a+log b 2 ) > 2 15 y = t n (t > 0) 16 2 < = t < = 2 1 (1) y = 1 4 x2 + x (2) y = x 2 2x, y = 2x 2 6x + 3 (3) y = x 2 8x + 15 (4) y = 1 4 x2 x 8 (5) y = x 2 4x + 3, y = 1 9 x2 4 3 x + 3 2 (1) a +, b, c, b 2 4ac + (2) a, b +, c, b 2 4ac + 3 (1) 4 < a 295

(2) 12 < a < 3 (3) a < 12 4 (1) D > 0, 1 <, f(1) > 0 (2) D > 0, < 1, f( 1) > 0 (3) f(2) < 0 (4) D > 0, 1 < < 3, f(1) > 0, f(3) > 0 (5) f(1) f(3) < 0 (6) f(1) f(3) < 0 D = 0, 1 < < 3 (7) f(2) < 0, f(5) < 0 (8) f(1) > 0, f(2) < 0, f(3) > 0 (9) f(1) > 0, f(2) < 0, f(5) < 0, f(6) > 0 5 (a 1) 2 + b 2 = 1 (b < 0) 2 < a y O 2 x 6 y(x + y + 1) < = 0 0 < = x 2 < = 1 y < = 0, x + y + 1 > = 0, x 2 4y > = 0 y 1 2 O x y > = x, y < = x y < = x, y > = x y > =, y > = x, y < = 1 2 x2 + 1 2 296

y 1 1 O 1 x 7 (m, n) = (3, 3) x = 2± 2 2 8 (a + b)(2a + b + 1) < = 0 2 < = a < = 0, b > = 1, b > = a, b < = 1 4 a2 a b 3 2 O a 297

1 0 < = a < = 1 g(a) = 1 1 < a < 1+ 3 = 2 g(a) = a 2 + 2a 1+ 3 2 < a g(a) = a 2 1 y 1 O 1+ 3 2 x 2 (1) x = 2 4a + 5 x = 0 1 (2) x = 2 4a + 5 x = a a 2 + 1 (3) x = 0, 2 1 x = 1 0 (4) x = 0 1 x = a a 2 + 1 (5) x = 0 1 x = 2 4a + 5 3 (1) (2) a < 1 (3) a > 1 (4) 4 (1) a = 3 32 55 64 (2) a = 1 14 5 x + y x = y = 3 6 x = 0, y = 2 2 x 2 + y 2 x = y = 3 18 x = 4 5, y = 8 5 16 5 2x + y 6 2 x 2 + y 2 45 4 16 5 6 (1) 1 2 (2) 2 2 (3) 3 3 4 7 5 8 2 9 (1) t = 2 3 n 298

(2) t = 1 2 2 t = 19 52 55 52 10 1 5 1 15 1 121 2 0.3 6 3 (1) n(n + 1)(n + 2) + (n 1)n(n + 1) 6n 2 (2) m 3 n mn mn 3 + mn = nm(m 1)(m + 1) mn(n 1)(n + 1) (3) n(n + 1)(n + 2)(n + 3) 4(n 1)n(n + 1) 4 (1) n = 7k+2 ( ) = 7(7k 2 +5k+1) n = 7k+4 ( ) = 7(7k 2 +9k+3) (2) n 7 n = (n 1)n(n + 1)(n 2 + n + 1)(n 2 n + 1) (n 1)n(n + 1) n = 7k n n = 7k + 1 (n 1) n = 7k + 2, 7k + 4 (n 2 + n + 1) n = 7k + 3, 7k + 5 (n 2 n + 1) n = 7k + 6 (n + 1) n 7 n 42 n 9 n 3 = n 3 (n+1)(n 1)(n 2 +n+1)(n 2 n+1) n = 3k, n = 3k +1, n = 3k + 2 9 5 (1) (x, y) = (5, 17), (6, 10), (11, 5), (18, 4), (3, 11), (2, 4), ( 3, 1), ( 10, 2) (2) (x, y) = ( 3, 0) 6 (1) (x, y, z) = (2, 3, 6), (2, 4, 4), (3, 3, 3) (2) (x, y, z) = (2, 3, 7) 41 42 7 (1) (x, y) = (2, 9), (3, 4) (2) (x, y, z) = (2, 3, 5) A < B < C π 3 < A 0 < A < π tan B+tan C 3 tan A = 1 tan (B + C) = 1 tan B tan C (tan B 1)(tan C 1) = 2 tan B = 2, tan C = 3 8 (1) k = 4 (2) (x, y) = (2, 1), (2, 3) B + C = 3 4 π 9 (1) 2 2 = q p, (p, q ) 2p2 = q 2 q q = 2r 2p 2 = 4r 2, p, p 2 = 2r 2 p q 2 299

(2) 3 + 2 2 3 + 2 2 = q p, (p, q ) 2 = q 3p 2p (1) 3 2 3 2 = q p, (p, q ) 2p3 = q 3 q q = 2r 2p 3 = 8r 3, p, p 3 = 4r 3 p q 3 2 (2) 2 + 3 2 + 3 = q p, (p, q ) 6 = q2 5p 2 2p 2 (3) log 2 3 = q p log 2 3 = q p, (p, q ) p log 2 3 = q log 2 2 3 p = 2 q log 2 3 (4) tan 1 tan 1 + 1 2 tan 1 = 1 tan 2 1 tan 2 tan 30 10 (1) 2 1 2 (2) 1+ 5 2, 1 + 2 11 n = 10k + l (l = 1, 2,, 9) 3 12 n = 3k, 3k + 1, 3k + 2 n 2 + 2 n = 3k 3k k = 1 n = 3 n = 3k, 3k + 1, 3k + 2 n = 3k + 1 n + 2 n = 3k + 2 n + 4 n = 3k 3k k = 1 n = 3 13 (1) a a = 2p a = 2p b 3 +2c 3 +4p 3 = 2bcp 14 (1) 0 15 15 213 b, c (2) a, b, c a = b = c = 0 (2) 4 16 pn 1 p 1 (3) n 1 n 3 17 (x, y) = (3m + 2, 5m + 2) 300

18 (x, y) = (13m 6, 212m + 98) 19 (1) (x, y) = (11m + 6, 14m + 7) (2) (x, y) = ( 5, 7) (3) 2(7x 6y) = 7 20 2x+3y = 17k (x, y) (x, y) = (3m+34k, 2m 17k) 9x+5y 9x+5y = 9(3m+34k)+5( 2m 17k) = 17(m+13k) 9x+5y 17 (x, y) = (3, 2), (37, 19) 21 1, 2, 4, 7 22 1 2 29 23 f(1) = a + b = m, f(2) = 4a + 2b = 2l m, l a = l m, b = 2m l f(x) = (l m)x 2 + (2m l)x f(x) n f(n) 24 (1) a = ±3 (2) a = 3 2x 3 + 3x 2 + x = x(x + 1)(x + 2) + (x 1)x(x + 1) f(x) 25 f( 1) = a + b c + d, f(0) = d, f(1) = a + b + c + d x = 3k 1 f(3k 1) = 3(9ak 3 9ak 2 + 3ak + 3bk 2 2bk + ck) a + b c + d 3(9ak 3 9ak 2 + 3ak + 3bk 2 2bk + ck) a + b c + d x = 3k, 3k + 1 26 (1) f(x) ±c (2) k (> 1) k f(1), f(2),, f(k) k x = mk + l m, l (0 < = l < = k 1) f(x) = 0 f(x) = 0 f(x) = 0 y (360, 361) y = x 27 x (1, 0) 180 181 (360, 361) y = x 301

28 l x = l l x+1 l x 1, l x, l x+1 l < k 2k l > k k 29 (1) p C k = p (p 1) (p k+2) (p k+1) k (k 1) 2 1 p (p 1) (p k + 2) (p k + 1) k k (k 1) 2 1 pc k p C k p p p C k p (2) (m + 1) p = m p + p C 1 m p 1 + p C 2 m p 2 + + p C p 1 m + 1 pc 1, pc 2, pc 3,, pc p 1 p (m + 1) p m p + 1 p (m + 1) p m p + 1 (mod p) (m + 1) p (m 1) p + 2 (mod p) (m + 1) p 1 p + m (mod p) (m + 1) p m + 1 (mod p) n p n (mod p) 1 2 2 7 2 3 (1) B = R AB = BC 4 (1) 1 (2) AB = AC A = 120 (2) 11 299 11 (3) 2 30 5 (1) 3 5 5 (2) 2 10 5 (3) 6 10 5 6 (1) 6 7 < a < 2 (2) 3 2 7 k < 0, 5 < k 0 k = 5 1 3 < k < 5, k = 0 2 k = 3 3 0 < k < 3 4 a < 6, 2 < a 0 a = 6, 2 1 6 < a < 2 27, 0 < a < 2 2 a = 2 27, 0 4 2 27 < a < 0 6 8 θ = 11 12 π 4 + 3 3 θ = 5 12 π 4 + 3 3 302

61 5 a = 5 3 2, b = 5 2 9 θ = π 6, p = 3 π 4 10 (1) 59 72 (2) α = 3 4 π, β = 7 4 π 11 π, 0, π 2 12 (1) 2π (2) 2π (3) π (4) 13 θ = 3 2 π 1 2 1 (1) 2 1 6 (2) 3 16 (3) 125 8 (4) a 2 b 2 (5) 1 (6) b 5 6 (7) 4 (8) 4 (9) 6 (10) 3 2 (1) 0 (2) 0.3010 (3) 0.4771 (4) 0.6020 (5) 0.6990 (6) 0.7781 (7) 303

(8) 0.9030 (9) 0.9542 (10) 1 3 0.84 4 (1) log 2 3 = q p p, q 3p = 2 q (2) n p = 2 q n = 2 m 5 (1) 0 (2) ±2 6 5 7 35 a = 2, b = 3, c = 5 8 2 9 35 18 10 4 log 5 2 < 2 < 3 log 5 3 11 log a a b < log b b a < 1 2 < log b a < log a b 12 (1) 2 18 (2) 100, 10 13 (1) 1 < x < = 3 5, 0 < = x < 1 (2) 0 < x < = 8 2 15, 8 + 2 15 < = x 14 (1) log 10 2 10 > log 10 10 3 (2) log 10 7 < log 10 8 (3) log 10 13 3 > log 10 2 11 15 (1) 44 (2) 8 10 43 < 18 35 < 9 10 43 16 90 5 48 17 1 2 < a < 1 18 a = 2 6 r +1, b = 2r +3 r 6 r +1 304

19 x = y 3 20 1 < = a < 5 4 x = 1 4 3a a = 5 4 x = ±1 1 4 5 4 < a < = 2 x = 1 1 4 x = log 2 a a 2 + a y y = x 1 O 1 x y = 1 x 21 y y = 2x y = 1 2 x 2 1 0 1 2 x 1 (1) a n = 3n + 43 (2) S n = 3 2 n2 + 83 2 n (3) n = 14 287 2 (1) a n = 2 n (2) S n = 2 n+1 2 (3) 12 3 β α αβ = β 2 0 < β < 1 α < αβ < β 1 < β αβ < α < β (α, β) = ( 1 2, 1 4 ), ( 2, 4) 305

4 n = 5 d = 5 n = 10 d = 1 5 6 (n+2)(n+1) 2 (3n+2)(n+1) 2 3n 2 + 3n + 1 7 (1) a n = 3n 2 + 3n (2) n 3n+3 (3) 3 2 n+1 (n 2 n + 2) 12 8 (1) n(3n+1) 2 9 129 2 (2) n(3n + 2) (3) 23n+2 4 7 10 10 16 7 84 11 (1) n 3 + 7 2 n2 + 15 2 n (2) 7 3 n3 + 3 2 n2 + 1 6 n 12 (1) n(n 1)(n+1)(3n+2) 24 (2) n2 (n 1)(n+1) 2 6 13 n(n+1)(n+2) 6 14 (1) n n+1 (2) n 2n+1 (3) n(3n+5) 4(n+1)(n+2) (4) 1 + n + 1 (5) n(n+3) 4(n+1)(n+2) 15 (1) 2n 2 4n + 3 (2) 32 47 (3) (2n 1)(2n 2 2n + 1) (1) 3 2 n2 3 2 n + 1 (2) 3 2 n3 1 2 n 16 1 2 n2 + 1 2 n + 1 17 1 6 n3 + 5 6 n + 1 306

1 (1) a n = 3 4 n 1 1 (2) a n = 1 3 ( 1 2 )n 1 + 4 3 (3) a n = 5n 4 (4) a n = 1 2 n2 1 2 n + 1 (5) a n = 2 n 1 (6) a n = 1 2n+1 (7) a n = (n + 1 2 )2n (8) a n = 4 3 n 1 2 n (9) a n = 2n +( 1) n 1 3 (10) a n = n+5 2 (11) a n = 2 n 1 + n + 1 (12) a n = 2 1 2 ( 1 3 )n 1 + 1 2 (13) a n = 2 2n 1 1 (14) a n = 2 3 n 1 2 n 1 (15) a n = (2n + 1)3 n 2 (16) a n = 3n 1 2, b n = 3n +1 2 2 (1) a n a n 1 = 1 2 (a n 1 a n 2 ) 1 2 (2) a n = 1 3 (a 1 + 2a 2 ) 2 3 (a 2 a 1 )( 1 2 )n 1 (3) log 2 b n = 2 3 2 3 ( 1 2 )n 1 3 (1) a n+1 = 3a n + 5b n, b n+1 = a n + 3b n (2) c n = (3 5) n (3) a n = (3+ 5) n +(3 5) n 2, b n = (3+ 4 2 3 n 3 2 n 1 5 a n = 6 (n+1)(n+2) 6 a n = 0 (n = 1), a n = 2 n(n 1) (n > = 2) 5) n (3 5) n 2 5 7 (1) a n+1 = (n + 1)a n (2) a n = n! (3) 1 1 (n+1)! 8 1 4 ( 1 3 )n 1 + 1 4 a n = 1 3 ( 1 2 )n 1 + 1 3 307

P 1 (n) = 1 4, P 2(n) = 1 4 ( 1 3 )n + 1 4 9 (1) 5 18 (2) ( 2 9 )( 4 9 )n 1 + 1 2 10 (1) (B, D), (B, G), (D, B), (D, G), (E, B), (E, D), (E, G) (2) r n = ( 7 9 )n (3) p n+1 = 1 3 p n + 2 9 q n (4) 3 11 (1) b n = n+1 n+3 b n 1 (2) b n = (3) a n = n n+2 (4) 1 e 2 12b 1 (n+2)(n+3) 1 n 3 (n 1) 3 2 (1) n = 1 = 1 2 = 1 = 1 6 1 2 3 = 1 = (2) n = k 1 2 + 2 2 + 3 2 + k 2 = 1 6k(k + 1)(2k + 1) n = k + 1 1 2 + 2 2 + 3 2 + k 2 + (k + 1) 2 = 1 6 k(k + 1)(2k + 1) + (k + 1)2 = 1 6 (k + 1)(k + 2)(2k + 3) n = k + 1 (3) n 3 n k=1 k = 1 n n(n + 1), 2 k 2 = 1 n 6 n(n + 1)(2n + 1), k 3 = k=1 { 1 2 n(n + 1)}2 k = 4 k=1 4 [ ] e iθ = cos θ + i sin θ (cos θ + i sin θ) n = (e iθ ) n = e inθ = cos nθ + i sin nθ 5 n = 1 19 + 1 2 = 21 = 3 7 n = 2 361 1 32 = 329 = 7 47 7 (1) n = 1 = 21 = 3 7 7 308

(2) n = k 19 k + ( 1) k 1 2 4k 3 = 7m n = k + 1 19 k+1 + ( 1) k 2 4k+1 = 19 k+1 + (7m 19 k ) ( 1) 2 4 = 19 k (19 + 16) 2 4 7m = 19 k 35 2 4 7m = 7(5 19 k 2 4 m) n = k + 1 7 (3) n 7 6 (1) n = 2 = 1 + 1 2 2 = 5 4 = 13 8 1 2 + 1 8 = 5 4 > = (2) n = k 1 + 1 2 2 + 1 3 2 + + 1 k 2 > = 13 8 1 k + 1 2k 2 n = k + 1 1 + 1 2 + 1 2 3 + + 1 2 k + 1 > 2 (k+1) 2 = 13 8 1 k + 1 2k + 1 2 (k+1) 2 13 8 1 k + 1 2k + 1 2 (k+1) { 13 2 8 1 k+1 + 1 1 2(k+1) } = > 2 2k 2 (k+1) 2 = 0 13 8 1 k + 1 2k + 2 1 > (k+1) 2 = 13 8 1 k+1 + 1 2(k+1) 1+ 1 2 2 + 1 2 3 + + 1 2 k + 1 > 2 (k+1) 2 = 13 8 1 k+1 + 1 2(k+1) 2 n = k + 1 (3) n > = 2 n 7 (1) n = 1 = x + 1 x = t n = 2 = x 2 + 1 x 2 = (x + 1 x )2 2 = t 2 2 n = 1, 2 t n (2) n = k, k + 1 t k n = k, k + 1 x k + 1 = f x k k (t), x k+1 + 1 x k+1 k = f k+1 (t) f k (t) t n = k + 2 x k+2 + 1 x k+2 = (x k+1 + 1 x k+1 )(x + 1 x ) (xk + 1 x k ) = f k+1 (t) t f k (t) t k + 2 (3) n x n + 1 x n 8 (1) 720 (2) 280 (3) 240 (4) 481 t n 9 x 3 30k + 20 x 4 15k 2 + 60k + 15 x 4 k = 2 10 80 C k 5 80 k x k P k = 80 C k 5 80 k P k+1 P k > = 1 k = 13 k = 12 309

11 0, 5, 25, 80 12 (1) (a + b) n = n C 0 a n + n C 1 a n 1 b + n C 2 a n 2 b 2 + + n C r a n r b r +... + n C n b n 1 1 x = 1, y = 1 (2) (a + b) n = n C 0 a n + n C 1 a n 1 b + n C 2 a n 2 b 2 + + n C r a n r b r +... + n C n b n x = 1, y = 2 (3) (a + b) n = n C 0 a n + n C 1 a n 1 b + n C 2 a n 2 b 2 + + n C r a n r b r +... + n C n b n x = 1, y = 1 2 (x 5) 2 + (y 5) 2 = 18 y = kx 1 7 < = k < = 7 3 ABCD AB, AC, AD, BC, CD, DB E, F, G, H, I, J EI O EG IG = 0 4 G EI AP = 7 9 3 AB+4AC 7 BC 4 : 3 Q P AQ 7 : 2 2 : 3 : 4 5 a, 6, 1, 6, 8, 9, 1, 4, 9, 5, 2, 2, 7, 5, 9, 8 6 (1) AP = 3 8AB + 1 4AC (2) AQ = 3 5AB + 2 5AC 7 (1) 1 (2) OF = 1 8OA + 3 8OB (3) 1 (4) AP BP = ( OP OA) ( OP OB) = OP 2 ( OA + OB) OP + OA OB = 2 OA + OB cos θ OA + OB = 2 5 2 + 2 5 2 2 5 8 (1) AI = 1 3 b + 4 15 c (2) k = 4 3 (3) 3 7 2 9 4 OA + 5 OB 2 = 6 OC 2 OA OB = 1 8 OB 2 2 OA OB + OA 2 = 9 4 10 11 (1) AQ = 5 7 b + 2 7 c (2) AH = 1 2 b + 1 5 c AB = 3 2 AB 2 = OB OA 2 = 310

12 OP = k( 1 3 a + 1 2 b) OP OA = OP OB OP = 5 9 a + 5 6 b OP = 1 12 (5 a + 3 b), 1 4 (5 a + 3 b) 13 OA = a, OB = b, OC = c a = b = c = R, AD BC = ( OD OA) ( OC OG = a+ b+ c 3 OD = 3 OG OB) = ( b + c) ( c b) = c 2 b 2 = 0 BD AC = 0 D ABC OH = 3 OG 14 (1) AM = b + e + 1 2 d (2) AN = 2 5 b + 2 5 e + 1 5 d 15 (1) OR = 1 4 a + 1 8b + 1 8 c (2) AN = 1 2 a + 1 4b + 1 4 c (3) AN = 2 7 a + 1 7b 16 (1) KN = KM + KL (2) OP = s c, OQ = (1 t) a+t b OR = 1 t 2 a+ t 2b+ s 2 c KR = OR OK = t 2 a + t 2b + s 2 c = t KL + s KM 17 AT = 3 13 b + 3 13 c + 3 13 d 18 (1) p a = 4(1 s t), p b = 4s + t, p c = s + t (2) s = 2 9, t = 4 9 (3) BQ : QC = 2 : 1, AR : RC = 4 : 3 (4) 4 : 3 : 2 OH = 1 5 a + 3 5b + 3 5 c OH = 10 5 19 (1) 91 2 (2) 17 (3) 23 4 (4) OH = 23 OP + 32 OM 55 55 20 (1) AI = b + 1 3 d + 2 3 e (2) k = 9 7 21 OA BC a ( c b) = 0 a b = a c a b = a c = b c OAB = OBC 1 2 a 2 b 2 ( a b) 2 = 1 2 b 2 c 2 ( b c) 2 a = b a = b = c OAB = ABC 1 2 a 2 b 2 ( a b) 2 = 1 2 a = b = c a 2 = 2 a b cos b a 2 c a 2 {( b a) ( c a)} 2 a b = a c = b c AOB = 1 2 AOB = π 3 a = b, AOB = π 3 OAB 311

1 (1) P (cos θ, sin θ, 0), P ( cos θ, sin θ, 0) AP = (cos θ 1, sin θ 1, 1), AQ = ( cos θ 1, sin θ 1, 1) cos 1 P AQ = 3 sin 2θ P AQ π 3 π 4 (2) P AQ = 1 2 AP AQ sin θ = 2 sin 2θ 3 1 2 S( 3 2, 5 2, 3 2 ), T ( 1 2, 2, 0) ST = 14 2 3 (1) 6x + 3y + 2z 6 = 0 (2) 38 7 (3) 7 2 (4) 19 3 4 (x 1) 2 + (y 2) 2 + (z + 13 4 )2 = 425 16 5 E( 5, 3, 1) 6 (1) A (1, 2, 3) (2) f(a) = 2 < 5, f(b) = 4 < 5 (3) P ( 7 4, 5 4, 3 4 ) 11 7 (1) H( 4 3, 2 3, 1 3 ) (2) P (1, 1, 0), Q( 4 3, 2 3, 1 3 ) 8 (1) H( 1 3, 2 3, 5 3 ) (2) ( 2 3, 2 3, 2 3 ), r = 2 6 3 (3) P ( 2+2 3 3, 2 3, 2+2 3 3 ) 9 x + y z 3 = 0, x + y + 5z 9 = 0 10 x 2y z = 0, 2x y + z = 0 P (± 6, 2 ± 6, 6) 11 (1) y + 2z 3 = 0 (2) x2 3 + (y+1)2 4 = 1 (1) (u, v, r) = (5 cos θ, cos θ, ± 2 2 cos 2 θ) (2) Y = 10 cos θ 5+cos θ, Z = 10 2 sin θ 5+cos θ (3) 144 625 (Y + 5 12 )2 + 3 25 Z2 = 1 1 (1) 5(x 3 + 2x 2 ) 4 (3x 2 + 4x) 312

(2) (x + 3) 2 (x 2 + 1)(3x 2 + 4x + 3) 2 a = 3, b = 2, 3 3 365 4 (1) α + β = 2a, αβ = b (2) 4a 3 6ab + 2c 5 x 2 + 2x 4 a = n, b = n 1 6 4 3 4 3 64 81 4 3 2 2 3 8 3 9 π 7 f(x) = x 2 + 2x + 1, 6 5 x 39 25 2 3 x + 19 27, x3 + 2x 2 + 2x + 1 8 p 2 8p 4 = 0 p > = 3 8 p = 4 + 2 5, q 9 y + z = x, yz = x 2 3 x 3 + y 3 + z 3 = 3x 3 9x x = 1 6 x = 1 6 1 y = 7x 2, y = x 2 2 y = 0, y = 4x (1) y = (2 2a)x a 2 + 2a 1 (2) 2 3 a3 3 y = 2x 5 4 4 0 < a < 1 2, 1 < a < 3 2 5 a < 1, 5 27 < a a = 1, 5 27 1 < a < 5 27 (1) a < 9 a = 9, 7 < a a = 7 9 < a < 7 (2) a < 2, 5 2 < a a = 2, 5 2 2 < a < 5 2 (3) p > 1 4 p = 1 4 p < 1 4 6 p < 1 4 p = 1 4 p > 1 4 7 a < 0, 1 < a a = 0, 1 0 < a < 1 313

2 < a < 2 8 k = 12 x = 4, 1, 3 k = 12 x = 3, 1, 4 9 71+8 2 2 k = 8 x = 1, 2, 4 10 (1) (2) n n (1) (2) f(x) = x n nx + 1 f (x) = nx n 1 n = n(x n 1 1) x x = ±1 f(1) = 2 n < 0, f( 1) = n > 0, f(2) = 2 n 2n + 1 > 0, f( 2) = 2 n + 2n + 1 < 0 x n nx + 1 = 0 2 11 x < = 3, 3 < = x y < = x 4 2x 2 + 1 3 < x < = 0 y < = 8 3 x + 4 3 3 0 < = x < 3 y < 8 = 3 x + 4 3 3 y O x 1 0 < x < 1 f(x) = 0 x x = α 1, 1 f(x)dx > = 2 α 1 1 α 1 1 f(x)dx > = 1 2 (1) b = 0, c = 1 3 a (2) a = 3 2, b = 0, c = 1 2 1 f(x)dx = α f(x)dx + 1 α 1 f(x)dx > = f(x)dx 314

3 (1) 2 3 4 64 27 5 27 4 (2) 343 24 6 a = 5 2 (1) F (t) = 4 3 t3 t 2 + 1 3 (0 < = t < = 1), F (t) = t 2 1 3 (1 < = t) (2) t = 1 2 1 4 7 a > = 1 S = 5a 7 3 0 < = a < 1 S = 8 3 a3 3a + 3 a = 6 4 6 6 2 (1) f(0) = a > 0, f( 1 2 ) = a 1 < 0, f(1) = a + 1 > 0 f(x) 0 < x < 1 (2) β α f (x) dx = 1 2 α 2a + 2 = a a = 2 3 12x 2 + 3dx + β 1 2 12x 2 3dx = (4α 3 3α) + (4β 3 3β) + 2 = 8 1 < x < 0 F (x) = 1 2 x2 + x + 1 2 x > = 0 F (x) = 1 2 x2 + x + 1 2 y 1 2 1 O 1 x 9 4 3 10 32 3 11 y = x 2 + 9 y = x 2 + 6 2 3 4 12 (1) C( a+b a+b 2, ab), D( 2, ( a+b 2 )2 ), M( a+b 2, a 2 +b 2 2 ) (2) 1 : 1 (3) 2 : 1 (1) 1 8 315

(2) 6 7 S 13 (1) y = 2ax a 2 (2) y = 1 2a x 1 16a 2 (3) S(a) = 1 6 (a + 1 4a )3 (4) a = 1 2 1 6 (5) T (a) = 1 12 (a + 1 4a )3 14 a = 0, b = 1 4, 1 12 1 (1) ( b 3a, 2b 3 27a bc 2 3a + d) (2) y = ax 3 + bx 2 + cx + d ( b 3a, 2b 3 27a 2 y = ax 3 + ( b2 3a + c)x a = A, (3) f( x) = f(x) (4) 1 : 2 2 (1) x = 2a (2) S 1 = 27 4 a4 (3) x = a, c = 4a 3 (4) x = 2a (5) S 2 = 27 4 a4 3 (1) 1 : 2 (2) 1 : 8 4 1 : 16 bc 3a + d) b2 3a + c = B y = Ax3 + Bx (1) x = t f(t + α) = f(t α) (4t + a)α 3 + (4t 3 + 3at 2 + 2bt+c)α = 0 α 4t+a = 0 4t 3 +3at 2 +2bt+c = 0 1 8 a3 1 2 ab + c = 0 (2) f(x) g(x) = x 2 + px + q, h(x) = x 2 + rx + s g(h(x)) g(h(x)) = (x 2 + rx + s) 2 + p(x 2 + rx + s) + q = x 4 + 2rx 3 + (r 2 + 2s + p)x 2 + (pr + 2rs)x + (ps + q + s) f(x) = x 4 + ax 3 + bx 2 + cx + d a = 2r, b = r 2 + 2s + p, c = pr + 2rs 1 8 a3 1 2 ab + c = 0 5 (1) f(0) = 0 (2) f (x) = x 2 + a (3) f(x) = 1 3 x3 + ax 6 (1) f n (1) = f n(1) = 0 316

(2) f 1 (x) = x 2 2x + 1, f 2 (x) = x 3 3x + 2 (3) 4n 2n+1 7 (1) (2) f n (x) = 4 3 (1 1 4 n )x 2 + 3(1 2 3 n )x 1 2 n 2 8 10 27, 46 27 9 P 1 (x) = x, P 2 (x) = x 2 1 3, P 3(x) = x 3 3 5 x 10 (1) f(x) = x, g(x) = 2x 5 2 (2) f(x) = 18x 12 11 (1) f(x) = 3 2 x2 2x + 1 (2) f(x) = 1 3 x3 1 2 x2 + 5 3 x + 1, g(x) = x2 x + 5 3 (3) a = 8, f(x) = 3x 2 + 8x 7, g(x) = x 2 + 4x + 1 12 (1) x = 0 1 4 x = 1 1 4 (2) i. 34 15 ii. x = a 2 13 f(x) = x 3 15 31 x2 12 35 x 18 31 f(x) = 3 2 x 1 2, C = 5 24 1 (1) 3x + 4y = 25 (2) 3x + 4y = 36 (3) 3x + y = 17 2 (1) y = 2, 4x 3y = 10 (2) 2x + y = 5, x + 2y = 5 3 (1) x 2y 2 = 0 (2) (2, 0), ( 2 5, 4 5 ) (3) (x 3 2 )2 + (y + 1) 2 = 5 4 4 (x + 9 2 )2 + (y + 3 2 )2 = 49 2 5 ( 2 3, 2 3 ) 2 2 3 (1) y = 3x + 6 (x < = 119, 1 < = x) (2) x = 1 5 6 ( 1 2, 1 2 ) 14 317

7 ( 1 5, 7 5 ) 2 30 5 8 6 33 9 (7, 8 ) (8, 4) 10 k = 9, 4 11 y = ± 1 15 (x + 1), y = ± 1 7 (3x 1) 12 a 1 5, ( 1, 0) 13 2ab l : y = b a x + 2b 14 (1) y = x 2 6x + 11 (2) y = x 2 6x 9 (3) 9x 2 24xy + 16y 2 + 34x 87y + 121 = 0 15 (1) D > 0 (2) 3a 2 2a2 2a+1 (3) a = 1 16 (1) 3 < 3 = k < 3 = 3 (2) OP OQ = 3 (3) P Q M M( 2 k 2 +1, 2k k 2 +1 ), R( 3 2, 3k 2 OP +OQ 2OR ( 2 ) = 2OR OM = 2 3 k 2 +1 2 2 17 (1) (0, 4), (4, 0) k 2 +1 = 6 ) OR(OP + OQ) = (2) 6 3 2 2 < a < 6+3 2 2 a = 6±3 2 2 a < 6 3 2 2, 6+3 2 2 < a 1 y = 3x 2 16x + 20 2 x 5y + 6 = 0, 5x + y 2 = 0 3 (1) OP OP = 1 (2) OP OP (3) ( 1 4, 1 8 ) 5 8 4 (x + 2 5 )2 + (y + 3 10 )2 > 1 4, (x 2 5 )2 + (y + 3 10 )2 < 1 4 (x + 2 5 )2 + (y + 3 10 )2 < 1 4, (x 2 5 )2 + (y + 3 10 )2 > 1 4 318