Asia Pacific Mathematical Olympiad for Primary Schools 2016 HANOI STAR - APMOPS 2016 Training - PreTest1 First Round 2 hours (150 marks) 24 Jan. 2016 Instructions to Participants Attempt as many questions as you can. Neither mathematical tables nor calculators may be used. Write your answers in the answer boxes. Marks are awarded for correct answers only. This question paper consists of 5 printed pages (including this page) Number of correct answers for Q1 to Q10 : Marks (x4): Number of correct answers for Q11 to Q20: Marks (x5): Number of correct answers for Q20 to Q30: Marks (x6):
1. The diagram below shows rectangle ABDE where C is the midpoint of side BD, and F is the midpoint of side AE. If AB = 10 and BD = 24, find the area of the shaded region. 2. Find the sum of all two-digit integers which are both prime and are 1 more than a multiple of 10. 3. Given that 1 1 1 1... 1, find the value of 1 1 1 1... 3 9 27 81 2 4 12 36 108 4. Trains go from town A to town B at regular intervals, all travelling at the same constant speed. A train going from town B to town A at the same constant speed along a parallel track meets the trains going in the opposite direction every 10 minutes. How often, in minutes, do the trains go from town A to town B? 5. If a, b and c donote the lengths ot the lines A, B and C in the picture, when which of the following statements is correct? A. a < b < c B. a < c < b C. b < a < c D. b < c < a E. c < b < a 6. In the number 2014, the last digit is greater than the sum of the other three digits. How many years ago did this last occur? 7. The length of the edges of the big regular hexagon is two times the length of the edges of the small regular hexagon. The small hexagon has an area of 4 cm 2. What is the area of the big hexagon?
8. How many digits are used to write the number 20 11? 9. Vivian wants to write the number 1000 as a sum of power of 3. At least how many powers of 3 does she need? 10. A two-digit even number is a multiple of 11. Multiplying the digits together will get a perfect cube and a perfect square. What is the number? 11. In the diagram below ABCDE is a regular pentagon, AG is perpendicular to CD, and BD intersects AG at F. Find the degree measure of AFB. 12. Six balls numbered 1, 2, 4, 8, 16 and 32 are placed in a bag. Ben removes four balls from the bag, adds up the number, writes the sum on a card and finally places the balls back into the bag. He repeats this process 15 times, where he obtains a different value each time. Find the sum of the values on the 15 cards. 13. The area of each of five circles is 133 cm 2. They are arranged in the form of cross inside a circle whose radius is three times as large. What is the total area, in cm 2, of the shaded parts in the diagram, taking 22? 7 14. Four studens Kate, Leonard, Michelle and Nelson, participated in an art competition. When asked about the results of the competition, the gave the following replies: - Kate: I am the first - Leonard: I am the last - Michelle: I am not the last - Nelson: I am neither the first nor the last If one of them lied, which of them emerged first in the art competition?
15. A circle is tangent to a line at A. From a point P on the circle, a line is drawn such that PN is perpendicular to AN. If PN= 9 and AN= 15, determine the radius of the circle. 16. Dick goes to school by bicycle, riding at the same constant speed every day. One day, 3 4 of the way to school, the bicycle breaks down, and he has to walk the rest of the way at a constant speed. If the amount of time Dick takes to go to school that day is twice the normal amount, how many times is his riding speed compared to his walking speed? 17. The perimeter of the big wheel of this bicycle is 4.2 m. The perimeter of its small wheel is 0.9 m. At a certain moment, the values of both wheels are at their lowest points. The bicycle starts rolling to the left. How many metres will the bicycle pass until both values are at the same time at their lowest point, for the first time again? 18. Paul put some rectangular paintings on the wall. For each picture, he put one nail into the wall 2.5m above the floor, and attached a 2m long string at the two upper corners. Which of the following pictures is closest to the floor (format: width in cm x height in cm) A. 60 x 40 B. 120 x 50 C. 120 x 90 D. 160.60 E. 160 x 100 19. Jane, Danielle and Hannah wanted to buy three identical hats. However, none of them had enough money to cover the price of one hat. Jane was short by a third of the price; Daniell by a fourth, and Hannah by a fifth. One week later, when there was a sale and the price of the hats was reduced by $9.40 per hat, the sisters combined their money and purchased all three hats, with no change left over. What was the price of one hat before the price reduction?
20. In the addition shown below A, B, C, and D are distinct digits. How many different values are possible for D? 21. Philip arranged the number 1, 2, 3,..., 11, 12 into six pairs so that the sum of the numbers in any pair is prime and no two of these primes are equal. Find the largest of these primes. 22. Each of 100 boxers has different strength, and in any match, the stronger boxer always wins. How many matches are needed to determine the strongest boxer and the second strongest one? 23. Pentagon ABCDE consists of a square ACDE and an equilateral triangle ABC that share the side AC. A circle centered at C has area 24. The intersection of the circle and the pentagon has half the area of the pentagon. Find the area of the pentagon. 24. The figure shows five circles A, B, C, D and E. They are to be painted, each in one color. Two circles joined by a line segment must have different colors. If five colors are available, how many different ways of painting are there? 25. On an island, frogs are always either green or blue. The number of blue frogs increased by 60% while the number of green frogs decreased by 60%. It turns out that the new ratio of blue frogs to green frogs is the same as the previous ratio in the opposite oder (green frogs to blue frogs). By what percentage did the overall number of frogs change? 26. Consider the set of all the 7-digit numbers that can be obtained using, for each number, all the digits 1, 2, 3,,7. List the numbers of the set in increasing order and split the list axactly at the middle into two parts of the same size. What is the last number of the first half?
27. Let ABC be the triangle such that AB = 6cm, AC = 8cm and BC = 10cm and M be the midpoint of BC. MADE is a square, and MD intersects AC at points F. Find the area of quadrilateral AFDE in cm 2. 28. Let [a] denote the largest integer not exceeding a. Find [S] if S = 1 2 + 2 2 2 + 3 2 3 + + 2015 2 2015 29. The sequence S1, S2, S3,, S10 has the property that every term beginning with the third is the sum of the previous two. That is, Sn = Sn 2 + Sn 1 for n 3. Suppose that S9 = 110 and S7 = 42. What is S4? 30. Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abc miles were displayed on the odometer, where abc is a 3-digit number with a 1 and a + b + c 7. At the end of the trip, the odometer showed cba miles. What is a 2 + b 2 + c 2? ------- THE END - Good luck to YOU ------
Name of Participant: Index No: Name of School: Asia Pacific Mathematical Olympiad for Primary Schools 2016 HANOI STAR - APMOPS 2016 Training - Pre Test1 - Answer Sheet No Answers No Answers No Answers Questions 1 to 10 each carries 4 marks Questions 11 to 20 each carries 5 marks Questions 21 to 30 each carries 6 marks 1 11 21 2 12 22 3 13 23 4 14 24 5 15 25 6 16 26 7 17 27 8 18 28 9 19 29 10 20 30 ------- THE END - Good luck to YOU ------