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

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1 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, 年 5 月 17 日 Time allowed: 2 hours 時限 :2 小時

2 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, contestants should write down steps of working, proofs, as well as the final answer, on Individual Contest - Answer Sheet (2). 在回答乙部分時, 參賽者應將計算過程證明及最後答案寫在個人項目答題紙 (2) 上 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. Attempt all questions in this paper. 本卷全部題目均需作答 5. 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 6. 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 7. The use of calculators is NOT allowed. 不可以使用計算機 8. 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 Individual (Junior) Page 2

3 The followings are some useful formulae and theorems: 以下是一些有用的公式及定理 : Menelaus Theorem 梅涅勞斯定理 Ceva s Theorem 塞瓦定理 Angle Bisector Theorem 角平分線定理 Fermat s Little Theorem 費馬小定理 Sum of consecutive perfect squares 連續平方數之和 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 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 can be divided 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 Individual (Junior) Page 3

4 Section A. Short Questions (100 marks) 甲部. 短題目 (100 分 ) Write down your answers only in the provided spaces in Answer Sheet (1). No working steps are required. 只需填寫答案在供應的答題紙 (1) 上, 不必詳列步驟 Six-mark Questions 六分題 Q1. Find the sum of the 7 marked angles in the following figure. 求右圖七個被標記的角的度數總和 Q2. Find the value of the following expression: 求以下算式的值 : Q3. Define. Evaluate. 定義求的值 Q4. As shown in the figure on the right, let ABCDEFGH be a cube of edge length 5. Let M, N be the mid-points of BC and EF respectively. Find the area of quadrilateral AMHN. 如右圖,ABCDEFGH 為邊長為 5 的正方體設 M, N 分別為 BC 和 EF 的中點求四邊形 AMHN 的面積 Q5. For a set of 5 not necessarily distinct positive integers, it is known that both their sum and their product are equal to. Find the product of all possible value(s) of. 已知 5 個不一定完全相異的正整數的和及積都是求的所有可能值之積 Q6. Find the number of positive even divisors of 10!. 請問 10! 有多少個正偶因數? Q7. In a grid paper, find the number of rectangles with area larger than or equal to 4. (The sides of all counted rectangles (including squares) should lie on the gridlines.) 在一的方格紙上, 求面積大於或等於 4 的長方形的數目 ( 所有被計算在內的長方形 ( 包括正方形 ) 的邊皆必定在方格線上 ) ISMC 2014 Individual (Junior) Page 4

5 Q8. A mathematics contest which consists of 30 questions, has a peculiar marking scheme. The first question a contestant answers correctly will be awarded 1 mark, and each question answered correctly afterwards grants 1 more mark than the previous correctly answered question. Each wrong answer will receive a 1 mark deduction from the contestant's total marks, and the base marks is 30. If the contestant opts not to answer a question, no marks will be awarded nor deducted for that question. If every contestant receives a different final mark in the competition, what is the maximum number of contestants? 有一個有 30 道題的數學競賽, 其評分方式非常怪異參賽者可於他答對的第一道題得 1 分, 以後每答對一道題所得的分數比於上一道答對的題目所得的分數多 1 分此外, 每答錯一道題倒扣 1 分, 底分為 30 分如某題未有作答, 則不會給予或倒扣任何分數假如每位參賽者最後得分都不相同, 請問最多有多少名參賽者? Q9. Kevin is walking from (0, 0) to (8, 8). He can either move 1 unit upwards (parallel to -axis) or 1 unit rightwards (parallel to -axis). He needs to pass through (1, 1), (3, 2) and (7, 7), find the number of possible paths. 振庭打算從 (0, 0) 走到 (8, 8) 他每一步都可以選擇上行一單位 ( 與 - 軸平衡 ) 或右行一單位 ( 與 - 軸平衡 ) 他需要經過 (1, 1), (3, 2) 和 (7, 7) 求可行路徑的數目 Q10. Consider a regular hexagon with side length of 6. Find the difference in area between the circumscribed circle and the inscribed circle. 考慮一邊長為 6 的正六角形求它的外接圓和內接圓的面積之差 Eight-mark Questions 八分題 Q11. Given that 12! = , find the value of the following expression: 已知 12! = , 求以下算式的值 : Q12. Let ABCD be a kite with side lengths of 5 and 12. Let E, F, G, H be the midpoints of AB, BC, CD, and DA respectively. Find the maximum area of quadrilateral EFGH. 設 ABCD 為一鷂形, 邊長分別為 5 和 12 設 E,F,G,H 分別為 AB,BC,CD 和 DA 的中點求四邊形 EFGH 的面積的最大值 ISMC 2014 Individual (Junior) Page 5

6 Q13. Given 3 consecutive positive odd integers. The sum of their squares is a multiple of Find the minimum value of. 給定三個連續正奇數, 他們的平方和為 1111 的倍數求的最小值 Q14. Isabella and Damian are taking turns to throw a fair dice, with Isabella going first. The first person who gets a cumulative total greater than 2 (the sum of dices rolls by the player so far) will win the game. Find the probability of Isabella winning the game. 芷瑜和迪文輪流擲骰子, 芷瑜先擲第一個先擲得總面值大於 2 的在這遊戲勝出 ( 總面值指那人從第一次擲骰子起所得之面值之總和 ), 求芷瑜勝出此遊戲的機率 Q15. Find the number of 7-digit positive integers that are divisible by 11 and their sum of digits equals to 45. 有多少個七位數字能被 11 整除, 而其數位數字之和又是 45? ***END OF SECTION A*** *** 甲部完 *** ISMC 2014 Individual (Junior) Page 6

7 Section B. Long Questions (100 marks) 乙部. 長題目 (100 分 ) Write your solutions to each question clearly on the answer sheet (2) provided. Correct answer with complete solution merits full marks. Even if you fail to solve the problem, partial credits may still be awarded under the following situations: 1. Correct answer(s) (Only account for about 20% of total score) 2. Incomplete solution(s) 3. Significant observation or creative idea 4. Discussion on non-trivial special case(s) 請在答題紙 (2) 上寫下詳盡解答正確答案及步驟可取得滿分, 若無法解答題目, 亦應寫下對該題目的想法評卷員會根據下列情況給予部分分數 : 1. 正確答案 ( 只佔約 20% 分數 ) 2. 未完成解答 3. 發現解答重點或有建設性想法 4. 對非顯然的特殊情況作出討論 Q1. Given that is positive. Evaluate. (25 marks) 已知是正數求的值 (25 分 ) Q2. Find the greatest common divisor (GCD) of 2016! and 2013! (25 marks) 求 2016! 和 2013! 2014 的最大公因數 (25 分 ) ISMC 2014 Individual (Junior) Page 7

8 Q3. Find the number of ways to label all the non-empty subsets of {1, 2,..., 6} with either 1 or 2, such that for any 2 non-intersecting sets (call them and ), the label of plus the label of minus the label of is always 0 or 2. (Definition for (Union): An element is in if and only if is in A or is in B.) (25 marks) 把 {1,2,..., 6} 所有的非空子集標籤為 1 或 2, 使得對於任意兩個沒有共同元素的子集, 標籤的數字之和減去標籤的數字永遠都是 0 或 2 請問有多少種方法標籤那些子集? ( 定義為的聯集 : 一元素在集內當且僅當在集內或在集內 ) (25 分 ) Q4. Let ABCD be an isosceles trapezium with AB // DC. Let E be the midpoint of AB, and F be a point on EC such that FDC = ECB. Prove that AD = AF. (Hint: Consider triangles ADF and EDC) (25 marks) 如下圖, 設 ABCD 為一等腰梯形, 當中 AB // DC 設 E 為 AB 的中點,F 為 EC 上的一點使得 FDC = ECB 證明 AD = AF ( 提示 : 考慮三角形 ADF 和 EDC) (25 分 ) *** 全卷完 *** ISMC 2014 Individual (Junior) Page 8

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

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, 2014 2014 年 5 月 17 日 Time allowed: 1 hour 時限 :1 小時 Instructions to contestants 參賽者須知 1. This paper consists