Download [PDF] APMOPS (SMOPS) 2010 Round 1 Questions with Answers Mathematical Olympiad

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[PDF] APMOPS (SMOPS) 2010 Round 1 Questions with Answers Mathematical Olympiad.pdf | Plain Text

Hwa Chong Institution SMOPS 17 th April 2010 Page 1 Asia Pacific Mathematical Olympiad for Primary Schools 2010 First Round 2 hours (150 marks) 1. Find the value of (1 − 1 2)× (1 − 1 3)× (1 − 1 4)× … × (1 − 1 98 )× (1 − 1 99 )× (1 − 1 100 ) (SMOPS 20 10 Q.1) 2. Find the value of 1 2+ 1 2×2+ 1 2×2×2+ ⋯ + 1 2×2×2×…×2 ⏟ 10 ������������ 2′������ . (SMOPS 20 10 Q.2) 3. Find the total number of ways to arrange 3 identical white balls and 3 identical black balls in a circle on a plane. The two layouts below are considered as one way of arrangement. (SMOPS 20 10 Q.3) 4. Given that one and only one of the following statements is true , which one is true ? (a ) All of the statements below are correct. (b ) None of the statements below is correct. (c ) One of the statements above is correct. (d ) All of the statements above are correct. (e ) None of the statements above is correct. (SMOPS 2010 Q.4) 5. Let n be a whole number greater than 1. It leaves a remainder of 1 when divided by any single digit whole number greater than 1. Find the smallest possible value of n. (SMOPS 20 10 Q.5)

Hwa Chong Institution SMOPS 17 th April 2010 Page 2 6. M and N are mid -points of the lines AD and BC respectively. Given that the area of ABCD is 2000cm 2, and the area of the shaded region ANCM = � cm 2, find the value of �. (SMOPS 2010 Q.6) 7. Your pocket money had previously been decreased by ________ %. To get back to the same amount of pocket money before the decrease, you need to have an increase of 25% . (SMOPS 20 10 Q.7) 8. A circle of diameter 2 cm rolls along the circumference of a circle of diameter 12 cm, without slipping, until it returns to its starti ng position. Given that the smaller circle turns �° abou t its centre, find the value of �. (SMOPS 20 10 Q.8) 9. Find the last digit of the number 2 × 2 × 2 × … × 2 ⏟ 859435 ������������ 2′������ (SMOPS 2010 Q.9) 10. Three bus services operate from the same bus interchange. The first bus service leaves at 24 minute intervals, the second at 30 minute intervals and the third at 36 minute intervals. All three services leave the bus interchange together at 0900. Find the number of minutes that has passed when they ne xt leave the interchange together. (SMOPS 20 10 Q.10)

Hwa Chong Institution SMOPS 17 th April 2010 Page 3 11. Twenty five boys position t hemselves in a 5 by 5 formation such that the distances between two adjacent boys in the same row or the same column are equal to 1 m. T he two dark circles indicate a p air of boys whose distance is exactly 5 m. given that there are n pairs whose distance apart are exactly 5m, find the value of n. (SMOPS 2010 Q.11) 12. When Albert begins walking up slope AB (1 km distance), across level ground BC (12 km distance), and down slope CD (3km distance), Daniel begins his journ ey in the opposite direction fro m D at the same time. Given that the speeds of both travelling up the slope, on level ground and down slope a re 2 km/h, 4 km/h and 5 km/h respectively, find the number of hours that has passed when they meet. (SMOPS 20 10 Q.12) 13. Find the value of 13+ 23+ 33+ 43+ ⋯ + 20 3+ 21 3. (SMOPS 20 10 Q.13) 14. Three identical circles have at most three points of contacts as shown below. Find the least number of identical circles required to have nine points of contact . (S MOPS 2010 Q.14) 15. A goat in a horizontal ground is tied to one end of a 14 m long rope. The other end of the rope is attached to a ring which is free to slide along a fixed 20 m long horizontal rail. If the maximum possible area that the goat can graze is x m2, find the value of x. [Ignore the dimension of the ring and take ������ = 22 7] (SMOPS 2010 Q.15) A B C D

Hwa Chong Institution SMOPS 17 th April 2010 Page 4 16. The 13 squares are to be filled with whole numbers. If the sum of any three adjacent numbers is 21, find the value of x. (SMOPS 20 10 Q.16) 4 9 x 17. A square is divided into three rectangles of widths �1 , �2 and �3, as shown. If �1+ �3= �2 and the areas of the shaded regions A , B and C are 8 cm 2, x cm 2 and 2 cm 2 respectively, find the value of x. (SMOPS 2010 Q.17) 18. Given that ������ = 1 12001 + 12002 + 12003 +⋯+ 12009 + 12010 , find the largest whole number smaller than S. (SMOPS 2010 Q.18) 19. The figure comprises five identical squares. �, �, � and � are vertices of the squares. �� cuts �� at ������ and angle �������� = �°, find the value of �. (SMOPS 20 10 Q.19) 20. Find the smallest whole number that is not a factor of 1× 2 × 3 × … × 21 × 22 × 23 . (SMOPS 2010 Q.20) �1 �2 �3 A B C A O C B D

Hwa Chong Institution SMOPS 17 th April 2010 Page 5 21. A square has its four vertices touching a circle and its four sides touching another smaller circle as shown below. If the ar ea of the larger circle is x times that of the smaller one, find the value of x. (SMOPS 2010 Q.21) 22. P, Q, R, S and T are equally spaced on a straight rod. If the rod is first rotated 180 ° about T, then 180 ° about S and finally 180 ° about P, which point’s position remains unchanged? (SMOPS 20 10 Q.22) 23. Given that the product of four different whole numbers is 10,000, find the greatest possible value of the sum of the four numbers. (SMOPS 20 10 Q.23) 24. An equilateral triangle PQR of side 32 cm has three equilateral triangles cut off from its corners to give rise to a hexagon ABCDEF . Another equilateral triangle LMN of side x cm gives rise to the same hexagon when subjected to the same treatment. If AB = 8 cm, BC = 15 cm, CD = 9 cm, DE = 10 cm, EF = 13 cm and FA = 11 cm, find the value of x. (SMOPS 20 10 Q.24) 25. The order of the following three numbers 3× 3× 3× … × 3 ⏟ 40 ������������ 3′������ 5× 5× 5× … × 5 ⏟ 30 ������������ 5′������ 7× 7× 7× … × 7 ⏟ 20 ������������ 7′������ from largest to smallest is _______. (1) A, B, C (2) A, C, B (3) B, C, A (4) B, A, C (5) C, A, B (6) C, B, A (SMOPS 20 10 Q.25) P Q R S T A C B

Hwa Chong Institution SMOPS 17 th April 2010 Page 6 26. B and D lie on AC and CE respectively and AD cuts BE at F. If B C = 2 AB , AF = 2FD , area of EFD = x cm 2 and area of BCDF = 1750 cm 2, find the value of x. (SMOPS 2010 Q.26) 27. In the diagram, AD = DC = CB , angle ADC = 110 °, angle DCB = 130 °, and angle ABC = �°, find the value of x. (SMOPS 20 10 Q.27) 28. The figure c omprises twelve equilateral triangles. Find the total number of trapezi a in the figure. Here we define a trapezium to be a 4 -sided figure with exactly one pair of parallel sides. (SMOPS 20 10 Q.28) A A B B C C D D E E F F A A B B C C D D

Hwa Chong Institution SMOPS 17 th April 2010 Page 7 29. The following 5 by 5 grid consists of 25 unit squares. Find the largest number of unit squares to be shaded so that each row, each column and each of the two main diagonal lines has at most 2 unit squares that are shaded. (SMOPS 2010 Q.29) 30. Find the value of x. (SMOPS 20 10 Q.30) 2 1 3 4 3 1 7 15 4 1 x 40

Hwa Chong Institution SMOPS 17 th April 2010 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 ) : _______ Answer s: SMOPS 2010 1 1 100 16 8 2 1023 1024 17 10 3 4 18 200 4 (e) 19 45 5 2521 20 29 6 1000 21 2 7 20 22 Q 8 2520 23 1257 9 8 24 34 10 360 25 (4) 11 8 26 250 12 21 2 27 55 13 53361 28 30 14 6 29 10 15 1176 30 13