2018IMAS Second Round Junior Division Problems

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Printed Name: Code: Score:

Junior Division Round 2

Time: 120 minutes

Instructions:

 Do not open the contest booklet until you are told to do so.

 Be sure that your name and code are written on the space provided above.  Round 2 of IMAS is composed of three parts; the total score is 100 marks.

 Questions 1 to 5 are given as a multiple-choice test. Each question has five possible options marked as A, B, C, D and E. Only one of these options is correct. After making your choice, fill in the appropriate letter in the space provided. Each correct answer is worth 4 marks. There is no penalty for an incorrect answer.

 Questions 6 to 13 are a short answer test. Only Arabic numerals are accepted; using other written text will not be honored or credited. Some questions have more than one answer, as such all answers are required to be written down in the space provided to obtain full marks. Each correct answer is worth 5 marks. There is no penalty for incorrect answers.

 Questions 14 and 15 require a detailed solution or process in which 20 marks are to be awarded to a completely written solution. Partial marks may be given to an incomplete presentation. There is no penalty for an incorrect answer.

 Use of electronic computing devices is not allowed.

 Only pencil, blue or black ball-pens may be used to write your solution or answer.  Diagrams are not drawn to scale. They are intended as aids only.

 After the contest the invigilator will collect the contest paper.

The following area is to be filled in by the judges; the contestants are not supposed to mark anything here.

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Junior Division Round 2

Questions 1 to 5, 4 marks each

1. Which of the following statement is false?

(A) If a divides b and k is an integer, then a divides kb. (B) If a divides b and b divides c, then a divides c.

(C) If a=bc, and b, c are positive integers, then a is divisible by b or c. (D) If b divides a and c divides a, then bc divides a.

(E) If p|bc, then p|b or p|c, where p is a prime number, b and c are integers.

Answer:

2. How many positive integers from 1 to 2019 can be expressed as n3−3n2 +2n, where n is an positive integer?

(A)11 (B)12 (C)13 (D)44 (E)45

Answer:

3. The perimeter of an isosceles triangle is known to be 32 cm and length of each side is an integer, in cm. How many different non-identical such triangles are there?

(A)5 (B)6 (C)7 (D)8 (E)9

Answer:

4. Given four distinct non-zero digits a, b, c and d, if ab+cd =dc+ba, then this expression is called a palindrome expression and the sum of the two numbers

ab+cd is called a palindrome sum. For example, 53 46+ =64 35+ =99. What is the minimum possible value of a palindrome sum?

(A)22 (B)33 (C)44 (D)55 (E)99

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5. In the figure below, the area of the trapezoid ABCD is 100 cm2, the area of parallelogram ABEF is 40 cm2 and CD=10 cm. What is the length, in cm, of AB?

(A)2 (B)2.5 (C)4 (D)5 (E)10

Answer:

Questions 6 to 13, 5 marks each

6. Four identical chess pieces are to be placed into a 4 4× chess board that is colored black and white alternately, as shown in the figure below. You can place at most one chess piece on each square. All chess pieces must be placed in squares of the same color and no two pieces are on the same row or on the same column. In how many different ways can the chess pieces be placed?

Answer: ways

7. It is known that for any 1

2 x≠ ± , then 242 4 1 1 4 1 2 2 a b x x x x + + = − + − . What is the value of a+b? Answer: J 2 A C B F D E

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8. In the figure below, ABCD is a square and the point G lies on the side CD. Now, flip triangle BCG along BG to get a new triangle BEG. If CBG= °32 , then what is the size, in degrees, of ∠DAE?

Answer: °

9. How many ordered triples (a, b, c) of integers that satisfy the equation |ab| |+ bc| |+ ca|=9?

Answer: triples

10. If x+ =y 4z−1, y+ =z 4x−1 and z+ =x 4y−1, where x, y and z are real numbers, then what is value of x+ +y z?

Answer:

11. Place 9 distinct positive integers into each of the unit squares of the 3 3× square below, with one number in each unit square, such that the sum of the numbers in every 2 2× square is 50. What is the minimum possible value of sum of the 9 integers?

Answer:

12. The three side lengths of an acute triangle are consecutive integers, in cm, and it is known that one altitude on one side is 12 cm. What is the area, in cm2, of this triangle? Answer: cm2 J 3 A C B G D E

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13. Arrange all positive integers less than 30 and not divisible by 3 in an increasing order, and compute the sum of the reciprocals of product of every three consecutive numbers, that is 1 1 1

1 2 4 2 4 5 26 28 29

S = + + +

× × × × L × × . Now, if

we reduce S into its simplest form, then what would be the value of the numerator?

Answer:

Questions 14 to 15, 20 marks each

(Detailed solutions are needed for these two problems)

14. In the figure below, a convex quadrilateral ABCD is inscribed in circle O. Points

E and F are on segments AB and AD respectively such that BE=CD and

DF =BC. If point M is the midpoint of EF, then prove that BMDM.

J 4 E C M A F B D

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15. A robot can generate a set of digit codes according to user’s reasonable instructions.Wayne gives out the following commands:

(1) Each code is a four-digit number (nonzero for the left-most digit). (2) Every two codes in the set have identical digits at no more than two

corresponding positions.

Find the maximum number of codes in a set the robot can generate.

Answer: codes

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