Singapore Mathematical Olympiad for Primary Schools Part A (70 marks (Q1 Q20, 2 marks each; Q30, Q21 3 marks each); time allowed 1.5 h) –
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1. Find the value of 1994 + 199.4 + 19.94 + 1.994 2. Find the value of 1 ÷ 0.1
×
0.01 ÷ 0.001
3. Let N be a 4-digit number. If N + 25 is a multiple of 8, find the smallest possible value of N. 4. What is the greatest 3-digit number in the number pattern: 1, 8, 15, 22, 29 36, ……
5. There are 30 coins which are made up of 20 ¢ coins and 50 ¢ coins. If the tota amount of money is $10.20, how many 20 ¢ coins and how many 50 ¢ coins are there? 6. One half of the rectangle below is divided into 5 equal parts. Then a triangle is drawn and shaded. What fraction of the whole rectangle is shaded?
7.
Find the total shaded area in the figure below:
8. The shaded part of the rectangle is a semi-circle. If the area of the rectangle is 20 cm ² , find the area of the semi-circle. (Take π = 3.14)
9. A rectangle field is 20 m long and 14 m wide. There is a path (as shown in the figure below) of width 2 m. Find the area of the path.
10. The following solids are made up of 6 unit cubes each. Which two of them are the same?
11. Use all the digits 1, 2, 3, 4 and 5 to form a 2-digit number and a 3-digit numbe such that the product of the two numbers is the greatest. What are the two numbers?
12. What is the ones digit in the value of
1 ² + 2 ² + 3 ² + 4 ² + ····· + 99 ² + 100 ² ?
13. If there are 4 Tuesdays and 4 Saturdays in January, what day of the week is 31 January?
14. In the 3 X 3 square, the sum of the numbers in any row, column or diagonal mus be the same. Find the value of a.
a
b
c
3
d
e
8
f
5
15. Three boys A, B and C went fishing. A caught 3 times as many fish as B. B caught 7 fewer fish than C. C caught 9 fewer fish than A. How many fish did A catch?
16. Six square cards each of side 10 cm are arranged as shown. Find the total area covered by the cards.
17. In the figure, A, B, C and D are the centres of the 4 circles. The radius of each circle is 10 cm. Find the total area of the shaded parts. (Take π = 3.14)
18. A parallelogram is divided into 4 smaller parallelograms A, B, C and D. The areas of A, B and C are 5 cm ² , 10 cm ² and 30 cm ² respectively. Find the area of D.
19. In how many ways can the word APPLE be formed by joining adjacent letters either vertically or horizontally? (one way is shown)
20. Two boys A and B each runs around a circular track of perimeter 1/5 km (as shown). They start from the point of contact of the two circles, O, at 8 a.m. and their speeds are 12 km/h and 18 km/h respectively. When will they first meet at the point O again?
21. Given that 4*2 = 14 5*3 = 22 3*5 = 4 7*18 = 31 Find the value of 6*9.
22. In the equation A × B = C, if A is increased by 12 and B remains unchanged, then C is increased by 60. If B is increased by 12 and A remains unchanged, then C is
increased by 144. Find the original value of C.
23. Ali took a number of mathematics tests. One of his scores was 79 marks and his average score was 88 marks. When the score 79 marks was changed to 97 marks his average score became 90 marks. How many tests did Ali take?
24. Five pupils A, B, C, D and E stand in a row in a certain order. They hold 30 flags together. The pupils on the right of C hold 23 flags. The pupils on the left of B hold 18 flags. The pupils on the left of D hold 12 flags. The pupils on the left of E hold 22 flags. If E is the first person on the right and C is the first person on the left, how many flags does A hold?
25. 300 people stand in a row. Starting from the first person on the left, each person calls out a number in this pattern: 1, 2, 3, 1, 2, 3, 1, 2, 3, ····· Starting from the first person on the right, each person calls out a number in this pattern: 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, ······ How many people call out the number 2 twice? 26. A 3 × 2 rectangle can be covered by 2 × 1 rectangles in 3 different ways as shown below:
In How many different ways can the following figure be covered by 2 × 1 rectangles?
27. There are 9 squares on each face of a cube (as shown). At most how many such squares can be painted red if any two squares with a common side cannot both b painted red?
28. The figure can be folded to form one of the cubes given below. Which cube can it form?
29. 8 locks and 10 keys are mixed up. If each lock can be opened by only one of the keys and no two locks can be opened by the same key, what is the smallest number of tries required to ensure that the correct key for every lock is found?
30. Some black and white pieces are placed on a 5 × 5 chess board as shown. A, B, C, ····· and I are empty spaces on the board. The score of a piece is the number of empty spaces adjacent to it. For example, the white piece x has two empty spaces namely E and G adjacent to it, so its score is 2 points.
At which empty space must we place a black piece so that the total score of the black pieces is the same as the total score of the white pieces?
Part B (50 marks; time allowed: 75 min)
1. There is a 100-digit number in which each digit is “1”. Find the remainder when this number is divided by 7.
2. Four boys A, B, C and D are passing a ball among them. Boy A starts passing the ball. After 3 passes, the ball returns to him. For example, the following is one of the different ways that the ball can be passed from one boy to another: A → C → D → A How many different ways are there altogether?
3. When the 11 th day of a month is a Sunday, the day is called a Happy Sunday. What is the greatest number of Happy Sundays in a leap year? (1 leap year = 366 days)
4. There are 600 children in Group A and 30% of them are boys. There are 400 children in Group B and 60% of them are boys. After some children are transferred from Group B to Group A, 40% of the children in Group A and 60% of the children in Group B are boys. How many children are transferred from Group B to Group A?
5. 11 boys arranged in a circle to receive books. They are numbered (clockwise) 1 to 11.
Boy 1 receives the first book; Boy 3 receives the second book; Boy 6 receives the third book; Boy 8 receives the fourth book; Boy 11 receives the fifth book; and so on (that is, the boys receive books in this pattern: skip one, skip two, skip one; skip two; ······ ) (a) Find the least number of books needed to be given away if every boy receives
at least 1 book. (b) Find the least number of books needed to be given away if every boy receives the same number of books. (c) If a total of 1994 books are given away, which boy receives the last book?
