Download [PDF] APMOPS (SMOPS) 2013 Round 1 Questions with Answers Mathematical Olympiad

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[PDF] APMOPS (SMOPS) 2013 Round 1 Questions with Answers Mathematical Olympiad.pdf | Plain Text

Hwa Chong Institution SMOPS 13 th April 2013 Page 1 Asia Pacific Mathematical Olympiad for Primary Schools 2013 First Round 2 hours (150 marks) 1. Find the last 5 digit of the sum. 127354 + 27354 + 7354 + 354 + 54 + 4. (SMOPS 2013 Q.1) 2. Find the sum of all two -digit numbers whose units digit and ten s digit are both even. (SMOPS 2013 Q.2) 3. The diagram shows a semi -circle with centre O overlapped with a parallelogram ABCD . The diameter, AB , of the semi -circle is 12cm. Find the total area of the shaded regions in cm 2. (SMOPS 2013 Q.3) 4. Abel, Ben and Charlie took part in SMOPS 2012, which comprised 30 questions. They had correctly answered 26, 23 and 18 questions respectively. What is the least possible number of questions that were answered correctly by all 3 students? (SMOPS 2013 Q.4) 5. Find the value of 555 × 554555 − 554 × 555554 . (SMOPS 2013 Q.5) 6. The diagram shows a cube with side length 5 cm. If a rectangular tunnel with dimensions 2 cm by 3 cm is made in the middle of the cube, find the amount of increase in the total surface area of the resulting solid in cm 2. (SMOPS 2013 Q.6) A B C D O

Hwa Chong Institution SMOPS 13 th April 2013 Page 2 7. The average value of four whole numbers �,̅ �5̅̅̅, �17̅̅̅̅̅and �432̅̅̅̅̅̅̅, where a, b, c and d each represents the first digit of a number, is 1735. Find the value of �+ �+ �+ �. (SMOPS 2013 Q.7) 8. The diagram shows a regular octahedron, which is a solid composed of eight equilateral triangles. Which of the following patterns can be folded into a regular octahedron? (SMOPS 2013 Q.8) 9. The sum of 2 prime numbers is equal to 2013. Find the product of these two numbers. (SMOPS 2013 Q.9) 10. The figure shows a big square, which is divided into four identical rectangles and one small square. Given that the area of the small square is 16 cm 2, the area of each rectangle is 140 cm 2, find the width of each rectangle in cm . (SMOPS 2013 Q.10) 11. A 5 -digit number written in the form 24 ��� ̅̅̅̅̅̅̅̅ has the last three digits unknown. If this number is divisible by 3, 4 and 5 respectively, find the greatest possible value that ���̅̅̅̅̅ can take. (SMOPS 2013 Q.11) Pattern (3) Pattern (2) Pattern (1)

Hwa Chong Institution SMOPS 13 th April 2013 Page 3 12. In the diagram, an ant is moving from A to B. If the ant is only allowed to move to the right or upwards along the grid lines, how many different paths are there from A to B? (SMOPS 2013 Q.12) 13. The diagram shows a parallelogram ABCD . E is the midpoint of AD . F is the midpoint of EC . If the area of the triangle BFD is 9 cm 2, find the area of the parallelogram ABCD in cm 2. (SMOPS 2013 Q.13) 14. If we write 2013 1990 in the form � + 1 � + 1 ������+ 1 �+1 � where a, b, c, d and e are positive integers, what is the value of �+ �+ �+ �+ �? (SMOPS 2013 Q.14) 15. In the given diagram, find the value (in degrees) of ∠� + ∠� + ∠� + ∠� + ∠� + ∠� (SMOPS 2013 Q.15) A B C B A D E F

Hwa Chong Institution SMOPS 13 th April 2013 Page 4 16. In how many different ways can four children share 8 identical chocolates so that each child get at least one? (SMOPS 2013 Q.16) 17. In the given diagram, each circle contains a natural number and the diagram satisfies the following conditions: i. The number labell ed along each edge represents the difference between the numbers in the two circles joined by the edge. ii. The sum of the numbers in the 5 circles is equal to 1979. Find the number in circle A. (SMOPS 2013 Q.17) 18. A certain type of water bottle is sold at $10 in both Store A and Store B. Mrs. Lim would like to buy a few water bottles for a Children’s Home. Store A sells the water bottle with an offer of “Buy 5 Get 1 Free” (no free bottles for buying fewer than 5 wat er bottles); store B gives a 15% discount for customers who buy 4 or more water bottles. What is the least amount of money (in $) that Mrs. Lim needs to spend in order to get 14 water bottles? (SMOPS 2013 Q.18) 19. A square of side length 18 cm is inscribed i n a circle. Semi -circle are constructed on its sides, as shown in the diagram. Find the total area of the four shaded lunes in cm 2. (SMOPS 2013 Q.19) A 2 6 3 5 4

Hwa Chong Institution SMOPS 13 th April 2013 Page 5 20. Four teams participated in a soccer tournament. Each team played against all other teams once each. 3 points were awarded for a win, 1 point for a draw and 0 points for a loss. At the end of the tournament, the four teams have obtained 5, 1, x, 6 points respectively. Find the value of x. (SMOPS 2013 Q.20) 21. In the diagram, the area of triangle ABC is 40. Given that 2 BD = 3 CD and AE = DE , find the area of the shaded region. (SMOPS 2013 Q.21) 22. If integers are selected randomly from 1 to35, what is the minimum numbers of integers that need to be selected such that among the chosen numbers we can always find two integers whose difference is divisible by 7? (SMOPS 2013 Q.22) 23. A team of workers are sent to two construction sites A and B respectively. The amount of work to be done at construction site A is 50% more than that at construction site B. In the morning, the number of workers sent to construction site A is 3 times the number of workers sent to site B. In the afternoon, the ratio of workers sent to construction site A and B is 7: 5. By the end of the day, the work at con struction site A is fully complete d, while construction site B still requires 8 workers to work for another full day. Assuming the workers work that the same rate, find the total numbers of workers in this team. (SMOPS 2013 Q.23) 24. Find the total number of triangles in the figure below . (SMOPS 2013 Q.24) A B C D E F

Hwa Chong Institution SMOPS 13 th April 2013 Page 6 25. A car and a motorcycle started travelling towards each other at the same instant, from cities A and B respectively. 72 minutes later, they met along the road and continued to travel towards their destinations. Given that the speed of the car is 11 3 times t hat of the motorcycle, how many minutes after the car reached city B would the motorcycle reach city A? (SMOPS 2013 Q.25) 26. Given four prime numbers a, b, c, and d, if the product of �× �× �× � is the sum if 55 consecutive positive integers, find the small est possible value of �+ �+ �+ �. (SMOPS 2013 Q.26) 27. A particular brand of car tyre lasts 300 km on a front wheel or 450 km on a rear wheel. By interchanging the front and rear tyres, what is the greatest distance, in km, that can be travelled using a set of four tyres of this brand? (SMOPS 2013 Q.27) 28. In triangle ABC , AD and AE trisect angle CAB , BD and BE trisect angle CBA . If the ratio of angle C to angle D is 1: 2, find the value of angle E in degrees. (SMOPS 2013 Q.28)

Hwa Chong Institution SMOPS 13 th April 2013 Page 7 29. The sum of 10 positive integers, not necessarily distinct, is 1001. If d is the greatest common divisor of the 10 numbers, find the maximum possible value of d. (SMOPS 2013 Q.29) 30. How many different ways are there to select 2 distinct integers from {2000 ,2001 ,2002 ,⋯ ,2014 ,2015 } such that the product of the 2 numbers is divisible by 6? (Note: order is not important, choosing 2001 and 2002 is the same as choosing 2002 and 2001.) (SMOPS 2013 Q.30)

Hwa Chong Institution SMOPS 13 th April 2013 Page 8 Number of correct answers for Q1 to Q10 : _______ Marks ( ×4 ) : _______ Number of correct answers for Q11 to Q20 : _______ Marks ( ×5 ) : _______ Number of correct answers for Q21 to Q30 : _______ Marks ( ×6 ) : _______ Answers: SMOPS 2013 1 62474 11 960 21 15 2 1080 12 23 22 8 3 36 13 72 23 72 4 7 14 100 24 17 5 1109 15 360 25 42 6 38 16 35 26 32 7 24 17 393 27 360 8 2 18 118 28 135 9 4022 19 324 29 91 10 10 20 4 30 47