Inter-School Mathematics Contest 2014 Individual Event (Junior Section) Questions

Inter-School Mathematics Contest 2014 Individual Event (Junior Section) consists of 8 pages. (中文版：按此)

Instructions to Contestants
  This paper consists of 19 questions. The total score of the paper is 200.  For Section B, contestants should write down steps of working, proofs, as well as the final answer, on Individual Contest - Answer Sheet (2).  Contestants may write on the back of each sheet of paper if necessary. Extra answer sheets will be given on request.  Attempt all questions in this paper.  Unless otherwise stated, all numbers in this paper are in decimal system.  Unless otherwise stated, answers should be given in decimal system and in the simplest form. No approximation will be accepted.  The use of calculators is NOT allowed.  The diagrams in this paper are not necessarily drawn to scale.</li> </ol>

Section A. Short Questions
<ul>The full marks of this section is 100 marks.</li>Write down your answers only in the provided spaces in Answer Sheet (1).No working steps are required.</li></ul>

Six-mark Questions

 * Q1. Find the sum of the 7 marked angles in the following figure.
 * Q2. Find the value of the following expression:

$$\frac{2^3-1}{1}+\frac{4^3-1}{3}+\frac{6^3-1}{5}+...+\frac{26^3-1}{25}$$


 * Q3. Define A$$\Delta$$B=A+B-AB-1. Evaluate [(1$$\Delta$$2)$$\Delta$$(3$$\Delta$$4)]$$\Delta$$[(5$$\Delta$$6)$$\Delta$$(7$$\Delta$$8)].


 * 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.
 * Q5. For a set of 5 not necessarily distinct positive integers, it is known that both their sum and their product are equal to x. Find the product of all possible value(s) of x.
 * Q6. Find the number of positive even divisors of 10!.
 * Q7. In a 65 grid paper, find the number of rectangles with area larger than or equal to4. (Note: "Rectangles" includes squares)
 * 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 contestants 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?
 * Q9. Kevin is walking from (0,0) to (8,8). He can either move 1 unit upwards (parallel to y-axis) or 1 unit rightwards (parallel to x-axis). He needs to pass through (1,1),(3,2) and (7,7), find the number of possible paths.


 * Q10. Consider a regular hexagon with side length of 6. Find the difference in area between the circumscribed circle and the inscribed circle.

Eight-mark Questions

 * Q11. Given that 12!=479,001,600, find the value of the following expression:

$$\frac{4^2-2}{4!}+\frac{5^2-2}{5!}+\frac{6^2-2}{6!}+...+\frac{12^2-2}{12!}$$


 * 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.
 * Q13. Given 3 consecutive positive odd integers n, n+2, n+4. The sum of their squares is a multiple of 1111. Find the minimum value of n.
 * Q14. Isabella and Damian are talking 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.
 * Q15. Find the number of 7-digit positive integers that are divisible by 11 and their sum of digits equals to 45.

Section B. Long Questions
<ul>The full marks of this section is 100 marks.</li>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: <ol>Correct answer(s) (Only account for about 20% of total score)</li>Incomplete solution(s)</li>Significant observation or creative idea</li>Discussion on non-trivial special case(s)</li></ol></li></ul>


 * Q1. Given that x is positive. Evaluate $$x=2014+\frac{1}{2014+\frac{1}{2014+\frac{1}{2014+...}}}$$.

(25 marks)


 * Q2. Find the greatest common divisor (GCD) of 2016! and 2013!-2014.

(25 marks)


 * Q3. Find the number of ways to label allthe non-empty subsets of {1,2,...,6} with either 1 or 2, such that for any 2 non-intersecting sets (call them A and B), the label of A plus the label of B minus the label of AB is always 0 or 2.
 * (Definition for ∪ (Union): AN element x is in A∪B if and only if x is in A or x is in B.)

(25 marks)


 * 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)

Solutions

 * 按此