Math Olympiad in Taiwan (IMSO)

Short Answer: there are 20 questions, fill in the correct answers in the answer
sheet. Each correct answer is worth 2 points. Time limit: 60 minutes.

1. In the 5×5 square the numbers 1, 2, 3, 4 and 5 are arranged in such a way that every number occurs precisely once in each column. In the 5×5 square shown, what is the entry in the position marked with * ?

2. The length of the sides of a triangle PQR are PQ=5, QR=3 and RP=4. The bisectors of the angles P and Q meet at the point I. What is the area of the triangle PQI?

3. When 10^2005?2005 is express as a single number, what is the sum of the digits?

4. What is the value of x in the diagram?

5. What is the number of lines of symmetry in the plane of the diagram?

6. A large watermelon weighs 12 kg, with 97% of its weight being water. It is left to stand in the sun, and some of the water evaporates so that now only 90% of its weight is water. What does it now weigh?

7. In Figure, OX=OY=10 are radii of a circular quadrant. A semi-circle is drawn on XY as shown. T, S and C denote the resulting triangles, segment and crescent. What is the area of C ?

8. Five students sit for an exam which has a maximum score of 100. The average of the five scores achieved by the students in the exam was 89. What could the minimum score be gained?

9. N is a positive integer such that N and N + 97 are both perfect squares. What is the positive integer N ?

10. Alan has a stride 75 cm. If he travels by walking 5 steps forward and one step back, what is the least number of steps he needs to reach a spot 24 metres away?

11. If a, b, c and d are positive integers such that

What is the value of a+b+c+d ?

12. A six digit number is represented by abcdef , where a, b, c, d, e and f are its
digits. If this number is multiplied by 6, the result is defabc . What is this six digit number?

13. How many rectangles are there in this grid, where vertices are points of the grid and the edges are lines of the grid?

14. A three-digit number N leaves remainder 3 when divided by 7, remainder 5 when divided by 11 and remainder 8 when divided by 17. What is the number N ?

15. What is the ratio of the shaded square to that of the largest square shown in the diagram?

16. There are 500 unit cubes. As many of these cubes as needed are glued together to form the largest possible cube which looks solid from any point on the outside but is hollow inside. What is the side length of the largest cube?

17. Mr. Sun has a broken calculator. When just turned on, it displays 0. If the + key is pressed, it adds 35. If the x key is pressed, it subtracts 35. If the × key is pressed, it adds 91. If the ÷ key is pressed, it subtracts 91. The other keys do not function. Mr. Sun turns the calculator on. What is the number closest to 2005 that he can get using this calculator?

18. Three man and three children arrive at the river where there is a small boat that will hold one adult or two children. What is the minimum number of trips across the river in either direction to get the family across?

19. A foundation has allocated a certain amount of money for 1st, 2nd and 3rd prizes in a competition. The money is divided in the ratio of 3:2 where the larger amount is for the 1st prize and the smaller amount is divided again in the ratio of 3?2 for the 2nd and 3rd prizes respectively. It becomes known that the 3rd prize is $3300 less than the first prize. How much is the 2nd prize?


3 Responses to “Math Olympiad in Taiwan (IMSO)”

  1. maria on August 22nd, 2011 11:10 am

    they were very useful.

  2. dei wang chun on September 5th, 2012 2:12 pm

    saya tidak mengerti ! bisa pakai bahasa indonesia ! saya lahir di beijing {cina} ! tapi saya tidak bisa bahasa china ! hahahahaha (bercanda)

  3. stella on March 10th, 2013 2:57 pm

    as a teacher,it’s very useful for me

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