2017IMAS First Round Junior Division Problems

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When your teacher gives the signal, begin working on the problems.

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2017 JUNIOR DIVISION FIRST ROUND PAPER

Time allowed：75 minutes

INSTRUCTION AND INFORMATION

GENERAL

1. Do not open the booklet until told to do so by your teacher.

2. No calculators, slide rules, log tables, math stencils, mobile phones or other

calculating aids are permitted. Scribbling paper, graph paper, ruler and compasses are permitted, but are not essential.

3. Diagrams are NOT drawn to scale. They are intended only as aids.

4. There are 20 multiple-choice questions, each with 5 choices. Choose the most reasonable answer. The last 5 questions require whole number answers between 000 and 999 inclusive. The questions generally get harder as you work through the paper. There is no penalty for an incorrect response.

5. This is a mathematics assessment, not a test; do not expect to answer all questions. 6. Read the instructions on the answer sheet carefully. Ensure your name, school

name and school year are filled in. It is your responsibility that the Answer Sheet is correctly coded.

THE ANSWER SHEET

2. Record your answers on the reverse side of the Answer Sheet (not on the question paper) by FULLY filling in the circles which correspond to your choices.

3. Your Answer Sheet will be read by a machine. The machine will see all markings even if they are in the wrong places. So please be careful not to doodle or write anything extra on the Answer Sheet. If you want to change an answer or remove any marks, use a plastic eraser and be sure to remove all marks and smudges.

INTEGRITY OF THE COMPETITION

The IMAS reserves the right to re-examine students before deciding whether to grant official status to their scores.

2017 JUNIOR DIVISION FIRST ROUND PAPER

Questions 1-10, 3 marks each

1. What is the simplified value of 2 2016 2017

( 18)− −1 − −( 1) ?

（A） 20− （B） 18− （C）0 （D）16 （E）18

2. Which of the following numbers is the sum of four consecutive positive integers?

（A）2016 （B）2017 （C）2018 （D）2019 （E）2020

3. In a supermarket, 3 kg of pear costs $16.26, while 2 kg of apple costs $13.62. How much does 1 kg of apple cost more than 1 kg of pear?

（A）0.61 （B）1.39 （C）1.42 （D）1.81 （E）2.64

4. The value of the faction m

n increases by 1 when the numerator increases by

2017. Find the value of n.

（A）1 （B）2016 （C）2017 （D）2018 （E）Uncertain

5. In the figure below, triangle ABC is an isosceles triangle with AB= AC. Point D

is on AC with BD=BC. If ∠ABD =21° , what is the measure, in degrees,

of∠BAC?

（A）21 （B）38 （C）42 （D）46 （E）54

6. What is the sum of all the prime divisors in the final result of 23 + + +03 13 73

(repeated prime divisors are counted only once)?

（A）7 （B）12 （C）13 （D）16 （E）64

B

A

D

7. Let a, b, c, d, e and f are distinct digits such that the expression ab+cd =ef .

What is the least possible value of ef ?

（A）30 （B）34 （C）36 （D）39 （E）41

8. How many integers x satisfy 2x+ ≤1 8?

（A）3 （B）4 （C）6 （D）8 （E）9

9. In the figure below, AQBC is a convex quadrilateral with QA QB= and

60 C

∠ = °. Triangle QAB is folded along AB to form triangle PAB. It is also

known that ∠PBC= °30 and ∠PAC = °20 . What is the measure, in degrees, of

AQB

∠ ?

（A）100 （B）110 （C）120 （D）130 （E）140

10. If all the edges of a rectangle are integers, then which of the following CANNOT be the length of its diagonals?

（A）5 （B）6 （C） 41 （D） 53 （E）10

Questions 11-20, 4 marks each

11. The length of 2 altitudes on adjacent sides of a parallelogram are 2 cm and 3 cm respectively with perimeter of the parallelogram is 18 cm. What is the area, in

cm2, of the parallelogram?

（A）9.6 （B）10 （C）10.5 （D）10.8 （E）12

12. Three students are to participate in four games. Each student participates in at least one game and each game has exactly one student participating. In how many ways can this be done?

（A）12 （B）18 （C）24 （D）30 （E）36 J 2 B A Q P C

13. The sum of a, b and c is negative, while the product of these three numbers is

positive. If x a b c

a b c

= + + , what is the value of x2017 −2017x2 +36?

（A） 1982− （B） 1981− （C） 1980− （D）1980 （E）1982

14. Given two positive integers, 4

7 of the first is exactly 2

5 of the second. What is

the minimum sum of these two integers?

（A）10 （B）14 （C）15 （D）35 （E）17

15. Tom starts working at 9：00 in the morning and finishes at 5：00 in the afternoon. How many more degrees does the minute hand rotates than the hour hand does on the clock during this period?

（A）120 （B）1200 （C）1320 （D）2640 （E）2880

16. The price criteria of the subway ticket of a city is as follows: $2 for within 4 km, $1 more per 4 km for distances between 4 km and 12 km, $1 more per 6 km for distances over 12 km. It costs $8 to take subway from station A to station B. Which of the following is closest to the distance between A and B?

（A）12 km （B）18 km （C）24 km （D）36 km （E）48 km

17. Consider a 3-digit number abc, where a, b and c are distinct digits, so that their

sum is 7. How many such three-digit numbers are there?

（A）6 （B）12 （C）14 （D）18 （E）22

18. If positive numbers x and y satisfy x2 −y2 = . What is the simplified value of 2

2 2 2 2 x + y − y x − ? （A）2 （B）2 2 （C）4 （D）4 2 （E）Uncertain J 3 A B

19. In the figure below, ABCD is a quadrilateral where AB= cm, 64 BC = cm, 5

CD= cm, 3DA= cm and ∠BAD= °. Find the length, in cm, of 90 AC.

（A）5 （B）7 （C） 34 （D）3 5 （E）2 13

20. If 1

3

a = and 1

4

b= , find the numerical value of a3 +b3−a b2 −ab2.

（A） 7 1728 （B） 7 1718 （C） 5 1718 （D） 5 1728 （E） 5 1628

Questions 21-25, 6 marks each

21. Let a, b and c be positive integers satisfy a2 bc 19 b c

a

+ = + + . Find the sum of a,

b and c.

22. The operation「⊗」satisfies:

(i) For all x and y, x⊗ =y (x− ⊗1) (y− + + ; 1) x y

(ii) For all x, x⊗ =1 1.

Find the value of (3⊗ ⊗ . 3) 3

J 4

A

D

C B

23. In the figure below, ABCD and AFCE are congruent rectangles with 20

AB= AF = cm and AD= AE=50cm. Find the area, in cm2, of AGCH.

24. In the figure below, the side length of equilateral triangle ABC is 6 cm. Each side is divided into 6 equal segments and connects corresponding dividing points to get an equilateral network. Call a point "reachable" if it can be connected to A by a broken line of length 5 cm along the grid lines without passing any lattice point twice. For example, point D in the figure is reachable. Find the number of

reachable points in the figure.

25. The students in a research class are clustered into two groups: the morning and afternoon sessions. A student takes part in exactly one group in each session (the two groups in each session can be different and the number of students in each group can be different). Each group has at least one student and at most 8 students. Each student reports the number of students in the group he or she belongs to in two sessions. One finds that no two students report the same pair of numbers (with order, for example, (1, 4) and (4, 1) are different). What is maximum number of students in the class?

＊＊＊ J 5 B A _D C E F H G B A D C