2016IMAS First Round Junior Division Problems

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

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2016 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.

2016 JUNIOR DIVISION FIRST ROUND PAPER

Questions 1-10, 3 marks each

( 20)− +16 −15 ?

（A）−19 （B）11 （C）21 （D）51 （E）61

2. The table below summarizes the results of a test in a certain class. What is the

total score of this class?

Summary of the results of a test

No. of students The highest score The lowest score The average score

42 100 16 84.5

（A）672 （B）3528 （C）3549 （D）4200 （E）4872

3. A three-digit number is not divisible by 24. When divided by 24, the quotient is a

and the remainder is b. What is the minimum value of a+b?

（A）5 （B）6 （C）7 （D）8 （E）9

4. In the trapezium ABCD, AB is parallel to CD. E and F are points on AD and BC

respectively such that EF is also parallel to AB. The area, in cm2 , of triangles

BAF, CDF and BCE are 8, 7 and 18 respectively. What is the area, in cm2 , of

ABCD?

（A）30 （B）32 （C）33 （D）35 （E）36

5. What is the value of the negative number x which satisfies |x− =3 | | 3 | 1x + ?

（A）−2 （B）−1 （C） 2 3 − （D） 1 2 − （E） 1 4 − B A F E D _C

6. The radius of each wheel of Rick's bicycle is 25 cm. He rides to school at a constant speed and arrives after 10 minutes. During this time, each wheel makes 160 revolutions per minite. Of the following five distances, which is closest to that between Rick's home and school?

（A）1 km （B）1.5 km （C）1.8 km （D）2 km （E）2.5 km

7. How many two-digit numbers are there such that at least one digit is divisible by 3?

（A）48 （B）54 （C）60 （D）66 （E）80

8. The chart below shows the sale figures of a certain merchandise in 2014 and 2015 by the season. How many more items were sold in 2015 than in 2014?

（A）23 （B）48 （C）85 （D）90 （E）110

9. ABC is an equilateral triangle. D is a point inside such that BCD is a right

isosceles triangle. The altitude BE of ABC intersects CD at F. What is the measure, in degrees, of ∠CFE?

（A）75° （B）70° （C）65° （D）60° （E）55°

10. In how many ways can 36 be expressed as the sum of two prime numbers, the

first larger than the second?

（A）1 （B）2 （C）3 （D）4 （E）5 J 2 B A F E D C 1st Quarter

Sales charts of a merchandise

2nd Quarter 3rd Quarter 4th Quarter

2014 2015 Sales items 50 100 150 200 250 134 180 233 210 157 205 270 235

Questions 11-20, 4 marks each

11. Every student in a class is either in the mathematics club or the language club, and one third of them are in both. If there are 22 students in the language club, 4 less than the number of students in the mathematics club, how many students are there in this class?

（A）12 （B）18 （C）24 （D）30 （E）36

12. The average of a group of numbers is 5. A second group contains twice as many numbers and its average is 11. What is the average when the two groups are combined?

（A）6 （B）7 （C）8 （D）9 （E）10

13. What is the value of xy if x− +1 1− + =x y 2016?

（A）2015 （B）2016 （C） 1

2016 （D）1 （E）0

14. Each of A and B goes to the gymnasium 3 or 4 times a week. After n weeks, A

has been there 57 times while B has been there only 47 times. What is the value of n?

（A）15 （B）16 （C）17 （D）18 （E）19

15. D is a point on AB such thatAD =1 and BD=2. How many points C are there

in the plane such that both ACD and BCD are isosceles triangles?

（A）2 （B）4 （C）5 （D）6 （E）8

16. From a 5 5× square piece of paper, two 2 4× rectangles are cut off along the

grid lines. In how many different ways can this be done?

（A）6 （B）9 （C）12 （D）18 （E）24

17. The number a is 5 more than its reciprocal. What is the value of (a2 −1)2 −125a?

（A）5 （B）25 （C）125 _（D）1 21

2

+

（E）5 21 J 3

18. With each vertex of a parallelogram ABCD as centre, a circle is drawn. Exterior common tangents are then drawn, as shown in the diagram below. If the perimeter of ABCD is 36 cm and the radius of each circle is 2 cm, what is the maximum area, in cm2 , of the figure enclosed by the circular arcs and tangents?

（A）117+4π （B）144+4π （C）153 4+ π （D）144 12+ π （E）153 12+ π

19. What is the smallest positive integer with 12 positive divisors such that it is

relatively prime to (20163−2016)?

（A）7007 （B）9163 （C）26741 （D）39083 （E）52877

20. At most how many right triangles can be formed by five lines on the plane?

（A）4 （B）5 （C）6 （D）7 （E）8

Questions 21-25, 6 marks each

21. The International Article Number has 13 digits ABCDEFGHIJKLM. Here M is a

check digit. Let S = +A 3B+ +C 3D+ +E 3F + +G 3H + +I 3J + +K 3L. If S

is a multiple of 10, then M is chosen to be 0. Otherwise it is chosen to be 10

M = −t where t is the remainder obtained when S is divided by 10. The Code

for a certain Article Number is 6901020□09017. What is the missing digit?

J 4

B A

22. What is the largest three-digit number which can be expressed as the sum of the cubes of three different positive integers?

23. The diagram shows a quadrilateral ABCD with ∠CDA=150°. The bisector of

DAB

∠ is perpendicular to BC and the bisector of ∠ABC is perpendicular to

CD. What is the measure, in degrees, of ∠BCD?

24. Let a and b be positive real numbers such that a2 =b b( +1) and b2 = +a 1.

What is the value of 1 1

a +b?

25. Each blouse cost 40 dollars, each skirt 70 dollars and each pair of shoes 80

dollars. Fanny bought at least one item of each kind, and spent at most 800 dollars. A outfit consisted of one item of each kind, and two outfits were different if they differed in at least one item. At most how many different outfits could there be? ＊＊＊ J 5 B A D C