2015 中學高級卷 參考解法

允許學生個人、非營利性的圖書館或公立學校合理使用

本基金會網站所提供之各項試題及其解答。可直接下載

而不須申請。

重版、系統地複製或大量重製這些資料的任何部分，必

須獲得財團法人臺北市九章數學教育基金會的授權許

可。

申請此項授權請電郵

ccmp@seed.net.tw

Notice:

Individual students, nonprofit libraries, or schools are

permitted to make fair use of the papers and its

solutions. Republication, systematic copying, or

multiple reproduction of any part of this material is

permitted only under license from the Chiuchang

Mathematics Foundation.

Requests for such permission should be made by

1-10 3 1. 21× 2015 (A) 45 231 (B) 54 321 (C) 42 315 (D) 14 325 (E) 23 514 21× 2015 = 42315 : (C) 2. K = L + R2 _{L = 4 K = 85 R R} (A) 5 (B) 7 (C) 9 (D) 4 (E) 8 L = 4 K = 85 85 = 4 + R2 _R2 _{= 81 R} R =√81 = 9 : (C) 3. 5 3 $1000 (A) $325 (B) $300 (C) $250 (D) $375 (E) $500 5 + 3 = 8 $1000× 3 8 = $375 : (D) 4. x (A) 108◦ (B) 105◦ (C) 90◦ (D) 120◦ (E) 102◦ x 60◦ 90◦ 108◦ 258◦ x◦ = 360◦− 258◦ = 102◦ : (E)

5. 3−2− 2−3 (A) −1 (B) 0 (C) − 1 72 (D) 1 72 (E) 17 72 3−2− 2−3 = 1 9 − 1 8 = 8 72 − 9 72 =− 1 72 : (C) 6. ( 初級卷 17 中級卷 11 ) 中 80 cm 中 88 cm (A) 112 cm (B) 132 cm (C) 96 cm (D) 168 cm (E) 156 cm 1 x y y y x x 2y +x = 88 2x+y = 80 y−x = 88−80 = 8 y = x+8 y 2x + y = 80 2x + x + 8 = 80 3x = 72 x = 24 y = 32 2x + 2y = 112cm : (A) 2 88− 80 = 8cm x cm (x + 8) cm 2x + x + 8 = 80 3x = 72 x = 24 2x + 2(x + 8) = 2× 24 + 2 × 32 = 112 cm : (A) 3 w h 2w + h 2h + w 3w + 3h = 80 + 88 = 168 w + h = 168÷ 3 = 56 2(w + h) = 112 cm : (A)

7. ( 初級卷 22 中級卷 14 ) 6 (A) 1 6 (B) 5 12 (C) 1 4 (D) 1 3 (E) 1 2 6 中 6 中 3 2 4 36 中 X X X X X XX X X X X X X X X 1 1 2 2 3 3 4 4 5 5 6 6 6 15 36 = 5 12 : (B)

8. 中 (A) 2 9 (B) 1 8 (C) 1 5 (D) 1 7 (E) 1 6 1 16 中 2 2 16 = 1 8 : (B) 2 中 中 _±120◦ 6 3 8 1 8 : (B) 9. 20 4 中 (A) 26 (B) 40 (C) 38 (D) 22 (E) 32 1 x x + 20 x + 20 + 4 = 3(x + 4) x = 6 6 + 20 = 26 : (A) 2 4 4 20 4 2 4 20÷ 2 = 10 10× 3 = 30 10− 4 = 6 30− 4 = 26 : (A)

10. ( 中級卷 12 ) 8 11 500 km km (A) 120 (B) 6 (C) 1 (D) 24 (E) 60 8 8× 24 = 192 11500 192 ≈ 12000 200 = 60 km : (E) 11-20 4 11. A {0, 1, 2} B {3, 6, x} 中 x A 中 B 中 C C 5 x (A) 12 (B) 4 (C) 24 (D) 0 (E) 6 A 中 B 中 0 0 0 3 6 x 6 12 2x 9 C = {0, 3, 6, 12, x, 2x} C 5 (i) x_{∈ {0, 3, 6, 12}} 2x /∈ {0, 3, 6, 12} x = 12 2x = 24 (ii) 2x∈ {0, 3, 6, 12} x /∈ {0, 3, 6, 12} 2x∈ {0, 3, 6, 12} x = 0 3 6 : (A)

12. 1 1 1 2 + 1 3 + ₁ 1 4 + 1 5 (A) 5 16 (B) 3 14 (C) 60 77 (D) 45 154 (E) 70 66 1 1 1 2 + 1 3 + ₁ 1 4 + 1 5 = 1 1 5 6 + 1₉ 20 = ₆ 1 5 + 20 9 = ₁₅₄1 45 = 45 154 : (D) 13. L cm× 200 cm 8 cm× 4 cm 5 cm L (A) 253 (B) 155 (C) 400 (D) 250 (E) 158 L cm 200 cm 5 cm 200÷ 4 = 50 3 cm 3× 49 = 147 cm L = 147 + 8 = 155 : (B)

14. 級 25 中 11 15 16 級 15 16 15 16 (A) 7 25 (B) 2 5 (C) 4 25 (D) 1 5 (E) 6 25 1 15 16 5 10 15 6 4 11 14 : (C) 2 15 中 15 15÷ (1 + 2) = 5 16 15− 5 = 10 14 16 中 14− 10 = 4 16 4 25 : (C) 15. 20 cm 10 cm cm (A) 5 (B) 5√3− 5 (C) 10− 5√2 (D) 4−√10 (E) 5(2−√3) 10 cm 20 cm O A B M 5 10 O AB 中 M M B = 10÷ 2 = 5 cm OB = 20÷ 2 = 10 cm r = OM △MBO r2 _{= 10}2 − 52 _{= 75} r = √75 = 5√3 10− 5√3 = 5(2−√3)cm : (E)

16. E G G E F 360 F G (A) 120 (B) 40 (C) 80 (D) 60 (E) 20 e f g E F G G E F g f e e + f + g = 360 e = 2g G E g F 0.5g f = 1.5g e + f + g = 360 2g + 1.5g + g = 360 g = 360 4.5 = 80 f− g = 0.5g = 40 F 40 G : (B)

17. 級 (10 ) 中 5 1 2 3 4 5 .. . 1 1 49 中 (A) (B) (C) (D) (E) 1 2 3 4 5 6 7 1 2 3 4 5 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 5 3 1 6( 1 中 ) X 中 X 中 5 2 7 5 3 4( ) : (C) 18. x y (2x_{+ 1)(2}y_{− 1) = 2015 xy} (A) 36 (B) 40 (C) 12 (D) 30 (E) 24 2015 2015 = 5× 13 × 31 2015 2 1 中 5 13 31 31 2 1 2y − 1 = 31 = 25 − 1 2x + 1 = 5× 13 = 65 = 26 + 1 x = 6 y = 5 xy = 5× 6 = 30 : (D)

19. N 1 N N N = 55 55 + 11 = 66 N 42 (A) 3 (B) 4 (C) 1 (D) 0 (E) 2 1 N 42 = N +N 2 = 3N 2 N = 2 3 × 42 = 28 N N k ≤ 1 3N 42 = N + k ≤ 4 3N N ≥ 3 4×42 = 31.5 33 41 中 N 33 35 37 39 41 k 11 7 1 13 1 N + k 44 42 38 52 42 N 28 35 41 3 : (A) 2 42 = 41 + 1 = 40 + 2 = 39 + 3 =· · · = 28 + 14 N 41 40 39 38 37 36 35 34 33 32 31 30 29 28 N 1 20 13 19 1 18 7 17 11 16 1 15 1 14 X X X N 28 35 41 3 : (A)

20. P QRS ST U V P S ST Q U P V X Y XQ + Y U (A) SU (B) RV (C) U Q (D) P R (E) P V P R V T Q U S Y P V 中 W SW P V P R V T Q U S X Y W P Q = P S ∠XQP = 90◦ − ∠XP Q = ∠W P S ∠XP Q = 90◦ − ∠XQP = ∠W SP △P XQ △SW P QX = P W Y U = W V QX + Y U = P W + W V = P V : (E)

21-25 5 21. 1 (A) √3 (B) 6(√3−√2) (C) 1 2 (√ 3 +√2) (D) 1 2 (√ 6 +√2) (E) 2 A B AB 1 高 1 高 A B 1 1 1 1 AB = 2×1 2+ 2× √ 3 2 = 1 + √ 3 √1 2AB = √ 2 2 ( 1 +√3)= 1 2 (√ 2 +√6) : (D)

22. 3 5 3 5 15 1 3 5 1 15 · · · ( 中 ) ( 中 ) 2015 (A) 1 15 (B) 3 (C) 5 (D) 1 3 (E) 1 5 3 5 15 1₃ 5 ₁₅1 1₃ 1_{5 15}1 3 1₅ 15 3 5 12 2015 = 12× 167 + 11 2015 11 1 5 : (E)

23. P Q R S ∠SRQ = 60◦ RS = 4 RQ = 6 SP = x P Q = y x y (A) x = 4 y = 2 (B) x = y = 3 (C) x = y = 4 (D) x = 4 y = 3 (E) x = 5 y = 2 P Q R S y 6 4 x 60◦ 1 SQ SQ = d P Q R S y 6 4 x 60◦ d P QRS ∠SP Q = 180◦ − ∠SRQ = 120◦ :

x2 + y2− 2xy cos 120◦ = d2 = 42 + 62− 2 × 4 × 6 × cos 60◦

x2 + y2+ xy = 16 + 36− 24 = 28 (x + y)2− xy = 28 (1) 中 x = 4 y = 2 : (A) 2 P QRS ∠SP Q = 180◦ − ∠SRQ = 120◦ RQ T △SRT P Q R S T y 2 4 4 x 4 60◦ 120◦ 120◦ P QT S x = 4 y = 2 P QT S x = 2 y = 4 P QT S 中 x = 4 y = 2 : (A)

24. a1 < a2 < a3 < a4 < a5 < a6 < a7 中 100 (A) 3 (B) 7 (C) 4 (D) 6 (E) 1 1 bi = ai+1 ai 中 1 ≤ i ≤ 6 a7 = a1b1b2b3b4b5b6 < 100 中 bi 2 • a1 ≥ 2 a7 ≥ 27 = 128 a1 = 1 • bi 中 4 a7 ≥ 1 × 25× 4 = 128 bi ≤ 3 中 1≤ i ≤ 6 • bi 中 3 a7 ≥ 1 × 24× 32 = 144 (b1, b2, b3, b4, b5, b6) (2, 2, 2, 2, 2, 2) (2, 2, 2, 2, 2, 3) 7 b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 a6 a7 7 : (B) 2 a7 1 2 4 8 16 32 64 a7 = 64 2 中 3 a7 = 96 6 a7 = 96 ak+1÷ ak 3 中 2 a7 > 100 7 : (B)

25. 4 2 (A) 6 5 (B) 1 3( √ 19 + 1) (C) √ 21− 2 3 (D) 2 3( √ 22− 2) (E) 12 5 2x P Q Q 中 P 中 P x x √ 2 x O Q P Q 2x √ 2 x O P Q (OP )2 = (2 + 2x)2+ (√2 x)2 = 4 + 8x + 6x2 OP = 4 6x2+ 8x + 4 = 16 3x2+ 4x− 6 = 0 x = √ 22− 2 3 2x = 2 3( √ 22− 2) : (D)

26-30 000-999 26 6 27 7 28 8 29 9 30 10 26. ( 中級卷 27 ) 2015 n 1 3 + 1 n n 1 1 3 + 1 n = n + 3 3n = a b 中 b < n gcd_{{a, b} = 1} k = gcd(n + 3, 3n) k = 3n b > 3b b = 3 k n + 3 3n + 9 k 3n k 9 k > 3 k = 9 9a = n + 3 9b = 3n n = 9a− 3 b = 3a− 1 < n n 6 15 24 33 . . . n 1 3+ 1 n = 1 3+ 1 9a− 3 = 3a− 1 9a− 3 + 1 9a− 3 = 3a 9a− 3 = a 3a− 1 = a b. n n ≤ 2014 中 n = 9a− 3 a 9a− 3 ≤ 2015 =⇒ 9a ≤ 2018 =⇒ a ≤ 2242 9 =⇒ a ≤ 224 224 : (224) 2 n 3 3n n + 3 n n 3 n 3 6 9 12 15 18 21 24 27 · · · 1 3 + 1 n 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 · · · 1 2 2 5 3 8 · · · n 3+1 3 n 6 15 24 33 . . . n 9 3 3 中 2015 2013 2016 = 224× 9 3 n 6 = 1× 9 − 3 2013 = 224× 9 − 3 224 : (224)

27. 100 × 100 × 100 P QRST U V W 1× 1 × 1 Z P S P Z = 33 V Z 1× 1 × 1 P S R Q T U V Z V U SP 100× 100 1×√2 V Z 33× 100 33 100 V Z V Z 32 99 32 + 99 = 131 131 + 1 = 132 1× 1 × 1 : (132) 28. ( 中級卷 29 ) 中 中 0 3 5 6 9 10 12 15 2 2 3 • 2₁₅× 1 30 • 2₁₅× 2 60 • 3× 3 15 90 2 15× 30 + 4 15× 60 + 9 15 × 90 = 4 + 16 + 54 = 74 : (74)

29. 38× 32 ABCD P Q R S AB BC CD DA AP P B BQ QC CR RD DS SA P QRS P QRS D C B A R Q S 1 中 O P R = QS r = OP = OS ABCD 中 E F G H x y 4 中 2 2 D C B A R Q P S E G F H O x y r r D C B A R Q P S E G F H Q′ S′ 中 P Q′RS′ △P RS △P RS′ P R △P RS′ P R 高 △P RS′ △P RS P Q′RS′ P QRS 中 P QRS △EOP x2+ 162 = r2 △HOS y2+ 192 = r2 x2− y2 = 192− 162 = 3× 35 = 105 (x + y)(x− y) = 3 × 5 × 7 中 x + y < 35 (i) x + y = 21 x− y = 3 (ii) x + y = 15 x− y = 7 (i) 中 x = 13 y = 8 △AP S △CRQ 1₂× 6 × 24 = 72 △BQP △DSR 1 2 × 8 × 32 = 128 P QRS 38× 32 − 2 × 72− 2 × 128 = 816 (ii) 中 x = 11 y = 4 △AP S △CRQ 1₂× 8 × 20 = 80 △BQP △DSR 1 2 × 12 × 30 = 180 P QRS 38× 32 − 2 × 80− 2 × 180 = 696 P QRS 816 : (816)

2 P QRS D C B A R Q P S a 38− a 38− a a 32− b b 32− b b P QRS = 38× 32 − ab − (38 − a)(32 − b) = 38b + 32a− 2ab 1 ≤ a ≤ 19 △AP S △BQP a b = 32− b 38− a b2_{− 32b + a(38 − a) = 0 b} b = 32± √ 322− 4a(38 − a) 2 = 16±√256− a(38 − a) a = 1 . . . 9 256− a(38 − a) = (a − 19)2_{− 105} : a 1 2 3 4 5 6 7 8 9 10 . . . 19 (a− 19)2_{− 105 219 184 151 120 91 64 39 16 −5 < 0} X X a = 6 b = 16± 8 a = 8 b = 16± 4 (a, b) (6, 8) (6, 24) (8, 12) (8, 20) 400 816 520 696 P QRS 816 : (816)

30. S 0 2015 S 1 S 0 2015 S k + 2 中 k 0 ≤ k ≤ 2014 S 0 2015 k S ={0, 1, 2, . . . , k, 2015} S (0 + 1 + 2 +· · · + k) + 2015 k + 2 = 1 2k(k + 1) + 2015 k + 2 = 1 2 ( k + 2 + 4032 k + 2 ) −3 2. (⋆) 1 2 ( k + 2 + 4032 k + 2 ) ≥ √ (k + 2)× 4032 k + 2 ⇒ k + 2 + 4032 k + 2 ≥ 2 √ 4032 中 k + 2 = 4032_k+2 k + 2 =√4032 = 63.49 . . . k = 61.49 . . . 0 ≤ k ≤ 2014 k + 2 +4032_k+2 k k (⋆) k = 61 k = 62 k = 61 S 1₂(63 + 4032₆₃ )− 3₂ = 62 k = 62 S 1₂(64 + 4032₆₄ )− 3₂ = 62 S 62 : (62) 2 n 0 2015 Sn = {0, 1, 2, . . . , n − 2, 2015} 中 2≤ n ≤ 2015 mn Sn mn= 1 2 (n− 1)(n − 2) + 2015 n = n2_{− 3n + 4032} 2n Sn+1 = Sn−1∪ {n − 1} n− 1 < mn m2 m3 m4 · · · n− 1 = mn n > 0 n− 1 < mn 2n(n− 1) < 2nmn 2n2− 2n < n2− 3n + 4032 n2+ n− 4032 < 0 4032 = 26× 32× 7 = 63 × 64 (n− 63)(n + 64) < 0 −64 < n < 63 m2 m3 m4 · · · m63 = m64 = 62 : (62)