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MATHEMATICS SHORT ANSWER PROBLEMS

Name : Index Number :

Country :

14th International Mathematics and Science Olympiad

Singapore

21 November 2017

Instructions:

1. Write your name, country and index number on both the Question Booklet and Answer Sheet.

2. Write your Arabic Numerical answers only in the Answer Sheet.

3. Each question is worth 1 mark. There is no penalty for a wrong answer.

4. For problems involving more than one answer, marks are awarded only when ALL answers are correct.

5. There are 25 questions in this paper.

6. You have 60 minutes to complete this paper.

7. Use black or blue pen or pencil to write your answer.

Page 1

International Mathematics and Science Olympiad 2017 SHORT ANSWER PROBLEMS

(1) Find the smallest positive integer n such that the three numbers: 96 n − , n and 96 n + are positive prime numbers.

(2) The figure below shows five circles touching each other at points B, C, D and E. Points A and B are two different points on the leftmost circle. Points E and F are two different points on the rightmost circle. An ant wants to crawl from A to F along part of the circumference of the circles. If the ant is only allowed to crawl along any parts of the circumference of the circles at most once, how many possible paths can the ant take to crawl from A to F?

(3) Alex spent a total of $119 in 7 days. Each day, he spent $4 more than the previous day. How much did Alex spend on the 6th day? (4) A sequence of numbers is added in the following way: For example, for the sequence {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, … }, all the numbers in the sequence are first written as decimals as shown below and then added up together. 0.1 0.02 0.003 0.0004 0.00005 0.000006 0.0000007 0.00000008 0.000000009 0.0000000010 0.00000000011 0.000000000012 + ⁞ 0.123456790123…

What is the result when the sequence {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, … } is added up in the same way?

A

B E C D

F

Page 2

(5) A positive integer is lucky if each digit in its base-ten representation, starting from the leftmost digit, is not more than the digit on its right. For example, 79, 335 and 679 are lucky numbers but 41, 523 and 786 are not. A super-lucky number is a lucky number with 8 in its ones place and whose square is also lucky. Find the smallest super-lucky number.

(6) Harry, Larry and Parry were each given some marbles on Monday. On Tuesday, Harry gave Larry and Parry some marbles so that Larry and Parry each ended up with 4 times their original number of marbles. On Wednesday, Larry gave Harry and Parry some marbles so that Harry and Parry each ended up with 3 times the number of marbles they had on Tuesday. On Thursday, Parry gave Harry and Larry some marbles so that Harry and Larry each ended up with twice the number of marbles they had on Wednesday. If Harry, Larry and Parry ended up with 48 marbles each on Thursday, how many marbles did Parry have on Monday? (7) Form 9-digit numbers using each of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 exactly once. Let A and B be two such numbers with 8 B A = . Find the sum of the digits of the number A B + . (8) Straight lines are used to divide a square into identical regions as shown in the figure below. How many triangles are there in the figure?

(9) A sculptor wanted to carve a statue from a piece of marble. On the first week, the sculptor carved out 35% of the original …