Instructions to contestants 參賽者須知 1. This paper consists of 10 questions. Each question is worth 30 marks and the total score is 300. 全卷共有 10 題, 每題總分為

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1 Question-Answer Book 問題答題簿 Inter-School Mathematics Contest 2014 Group Event 2014 聯校數學比賽團體賽 May 17, 年 5 月 17 日 Time allowed: 1 hour 時限 :1 小時

2 Instructions to contestants 參賽者須知 1. This paper consists of 10 questions. Each question is worth 30 marks and the total score is 300. 全卷共有 10 題, 每題總分為 30 分, 全卷滿分為 Contestants should write down steps of working, proofs, as well as the final answer, in the space provided for each question. 參賽者應將計算過程證明及最後答案寫在每題的空白位置 3. Contestants may write on the back of each sheet of paper if necessary. Extra answer sheets will be given on request. 如有需要, 可在每張紙的背面書寫, 及向監考員索取附加答題紙 4. Contestants should write down their team code in the blank provided on the page for each question. 參賽者應在印有題目的每頁的橫線上寫上隊伍編號 5. Answer all questions. 全部題目均需作答 6. Unless otherwise stated, all numbers in this paper are in decimal system. 除非特別指明, 本卷內所有數字皆以十進制表示 7. Unless otherwise stated, answers should be given in exact values in decimal system and in the simplest form. No approximation will be accepted. 除非特別指明, 所有答案皆應以十進制的精確值表示, 並化至最簡不接受近似值 8. The use of calculators is NOT allowed. 不可以使用計算機 9. The diagrams in this paper are not necessarily drawn to scale. 本卷內的附圖未必依比例繪成 ISMC 2014 Group Event Page 2

3 The followings are some useful formulae and theorems: 以下是一些有用的公式及定理 : Sine Rule 正弦定理 Cosine Rule 餘弦定理 For any ABC, a, b, c are the opposite sides of A, B, C respectively, we have: 對於任意 ABC,a b c 分別為 A B C 的對邊, 則有 : c 2 = a 2 + b 2 2ab cos C Menelaus Theorem 梅涅勞斯定理 Ceva s Theorem 塞瓦定理 Angle Bisector Theorem 角平分線定理 Power Chord Theorem 圓冪定理 Power of P to the circle is: P 點對該圓的圓冪為 PO 2 r 2 = PA PB = PC PD Ptolemy s Theorem 托勒密定理 AB CD + AD BC = AC BD Heron s Formula 海倫公式 Binomial Expansion 二項式定理 ISMC 2014 Group Event Page 3

4 Arithmetic Series 等差數列之和 Geometric Series 等比數列之和 Compound Angle Formulae 複角公式 Fermat s Little Theorem 費馬小定理 AM-GM Inequality 均值不等式 a is the first term, d is the common difference, n is the number of terms a 為首項,d 為公差,n 為項數 a is the first term, d is the common ratio, n is the number of terms a 為首項,d 為公比,n 為項數 sin (A ± B) = sin A cos B ± sin B cos A cos (A ± B) = cos A cos B sin B sin A If p is a prime, then for any integer a, a p a is divisible by p. 若 p 為一質數, 對於所有整數 a, 皆有 a p a 能被 p 整除 For any list of n non-negative real numbers x1, x2,, xn, we have: 對任意 n 個非負實數 x1, x2,, xn, 則有 : Cauchy-Schwarz Inequality 柯西不等式 Sum of consecutive perfect squares 連續平方數之和 For any list of n real numbers x1, x2,, xn and y1, y2,, yn, we have: 對任意 n 個實數 x1, x2,, xn 及 y1, y2,, yn, 則有 : n is a natural number, we have: n 是自然數, 則有 : The followings are some common notations: 以下是一些常用的記號 : x R, y N x y (mod m) a b n! x is a real number, and y is a natural number. x 是實數,y 是自然數 x and y leave the same remainder when divided by m. x 和 y 除以 m 時的餘數相同 a is divisible by b. a 能被 b 整除 For any positive integers n, n! = 1 2 n. Moreover, 0! = 1. 對任何正整數 n,n! = 1 2 n 另外,0! = 1 For any positive integers n and r, where n r, we have: 對任何正整數 n 和 r, 其中 n r, 則有 : ISMC 2014 Group Event Page 4

5 Q1. The following figure shows the graph of a heart-shaped curve with equation in scale. Find the number of lattice points (where both and are integers) that lies on or within the area bounded by the curve. 心形方程在平面上的圖案依據比例如下圖所示求被此圖案包含在內的整數格點數目 ( 包括在方程上的整數格點 ) ISMC 2014 Group Event Page 5

6 Q2. Let [ ] denote the greatest integer not exceeding. For example: [ ] = 3, [2014] = 2014 and (a) [5.17] = 5. Evaluate the following sum: 設 [ ] 代表不超過的最大整數, 例如 [ ] = 3 [2014] = 2014 和 [5.17] = 5 計算以下算式的值 : (b) ISMC 2014 Group Event Page 6

7 Q3. Let. Given that S is a real number, evaluate S. 設已知 S 是實數, 求 S ISMC 2014 Group Event Page 7

8 Q4. Given 14 points on the circumference of the circle, Benny wants to draw 7 chords such that every point lies on exactly 1 chord. How many possible ways are there for him to draw the 7 chords if: (a) there are no additional restrictions; (b) an additional restriction is given: any 2 chords do not intersect each other. 給定圓上有 14 點, 本一想畫 7 條弦使得所有點皆必定在 1 條弦上請問他有多少種方法畫弦, 如果 : (a) 沒有增設任何附加限制 ; (b) 增設附加限制 : 任意 2 條弦並不相交 ISMC 2014 Group Event Page 8

9 Q5. As shown in the following figure, A is the centre of the larger circle. A smaller circle is constructed through A and it intersects the larger circle at B. The line passing through B and C is a tangent to the smaller circle. Also, BD is the angle bisector of ABC and D is a point lying on AC. Show that if D lies within the smaller circle, then ABC > 72. 如下圖所示,A 是大圓的圓心過 A 畫一小圓, 且此小圓與大圓於 B 相交過 B 和 C 的直線同時為小圓於 B 的切線此外,BD 為 ABC 的內角平分線,D 是 AC 上的一點如果 D 在小圓內, 試證 ABC > 72 ISMC 2014 Group Event Page 9

10 Q6. Given that > 0 and they are not necessarily distinct, find the number of 4-digit number such that and. 已知正整數 > 0, 而且它們不一定完全不同請問有多少個四位數滿足條件和? ISMC 2014 Group Event Page 10

11 Q7. Given that and are both non-negative integers satisfying the equation, find the sum of all possible values of. 已知和均為非負整數, 且滿足方程, 求的所有可能值之和 ISMC 2014 Group Event Page 11

12 Q8. Define as the product of all positive divisors of the positive integer. For example,. (a) Find the minimum possible positive integer such that there exists a positive integer satisfying. (b) For all positive integers, is always an integer? If yes, prove it. If not, give a counter example and explain briefly. 定義為正整數的所有正因數的積例如, (a) 求最小正整數使存在正整數使 (b) 對於任意正整數, 反例並解釋之必定是整數嗎? 如果它必定是整數, 試證之否則, 請舉一 ISMC 2014 Group Event Page 12

13 Q9. Let [ ] denote the greatest integer not exceeding and. For example: [5.17] = 5 and {5.17} = Evaluate. 設 [ ] 代表不超過的最大整數, 例如 [5.17] = 5 和 {5.17} = 0.17 求的值 ISMC 2014 Group Event Page 13

14 Q10. In ΔABC, K lies on AC and BK is the angle bisector of ABC as shown. L is the foot of the altitude from K to CB, and D is the foot of the altitude from C to AB. CD intersects KL and BK at N and M respectively. If the circumcircle of ΔKNB intersects the interior of AB at P, show that ΔKMP is isosceles. 如下圖所示,K 在 AC 上,BK 是 ABC 的內角平分線 L 是 K 至 CB 的垂足,D 是 C 至 AB 的垂足 CD 分別與 KL 和 BK 交於 N 和 M 如果 ΔKNB 的外接圓交 AB 於 P, 且 P 在 A 和 B 之間, 試證明 ΔKMP 為一等腰三角形 *** 全卷完 *** ISMC 2014 Group Event Page 14

Instructions to contestants 參賽者須知 1. This paper consists of 19 questions. The total score of the paper is 200. 全卷共有 19 題, 全卷滿分為 200 分 2. For Section B

Instructions to contestants 參賽者須知 1. This paper consists of 19 questions. The total score of the paper is 200. 全卷共有 19 題, 全卷滿分為 200 分 2. For Section B Questions Booklet 問題簿 Inter-School Mathematics Contest 2014 Individual Event (Junior Section) 2014 聯校數學比賽個人賽 ( 初級組 ) May 17, 2014 2014 年 5 月 17 日 Time allowed: 2 hours 時限 :2 小時 Instructions to contestants