Đề thi Toán quốc tế IMSO năm 2011

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Math Short Answer Problems IMSO 2011

1. The sum of the three smallest prime numbers and another prime number n is 2021. Find the value of n. 2. The average of x and y is 19. The average of a, b and c is 14. Find the average of x, y, a, b and c. 3. The figure is made up of semicircles of diameter 2 cm, 4 cm and 8 cm. What fraction of the figure is shaded?

4. The sum of the numbers A, B and C is 390. Given A is 3 times of B and A is one third of C, find the value of C.

5. What is the smallest number that can be expressed as the sum of two squares in two different ways? 6. How many rectangles are there in the 3 3 × grid shown?

7. Find the smallest positive integer with exactly 30 factors. 8. Find the value of

      + +       + + + −       + + +       + + 5 2 2011 2010 2010 2009 7 6 2011 2010 2010 2009 2 1 5 2 2011 2010 2010 2009 2 1 7 6 2011 2010 2010 2009

9. The number 119 is very amazing. When divided by 2, it leaves a remainder of 1. When divided by 3, it leaves a remainder of 2. When divided by 4, it leaves a remainder of 3. When divided by 5, it leaves a remainder of 4. When divided by 6, it leaves a remainder of 5. Find the smallest three-digit number, larger than 119, which has this property.

10. Pedro can finish a job in 14 minutes while his younger brother Juan can finish the same job in 7 minutes. How long will it take the two of them to finish the job together? 11. In the diagram shown, find the measure of

∠a + ∠b + ∠c + ∠d + ∠e .

12. The sum of the two digits of Emma’s age this year is 5. Seven years from now, her age will be 2 less than the reverse of the digits of her age this year. How old is Emma now? 13. Half of the punch in the bowl is pure mango juice. When 6 more cups of

pure mango juice are added to the bowl, 3

2 of the resulting punch in the

bowl is pure mango juice. How many cups of punch were in the bowl to start with? 14. Five test scores have a mean (average score) of 88, a median (middle score) of 89, and a mode (most frequent score) of 93. What is the sum of the two lowest test scores? 15. The area of a rectangle with integer sides is 2011 cm2. Find its largest possible perimeter, in cm. 16. In the grid shown, how many paths

are there from corner A to corner B that only have steps to the right or up and do not pass through neither C nor D? A

B

C

D

e

d c

b

a

17. Chords EF and GH are perpendicular to each other. A and B are centers of circle A and circle B respectively. Circle A and circle B are tangent to the chords, as shown in the diagram. If the radius of circle A is 12 and the radius of circle B is 5, find the length of line segment AB.

19. A square piece of paper has area 96 cm2. It is folded along the dotted line MN. The shaded area in the resulting figure is one-half of the total visible area. Find the length of MN.

20. What is the maximum number of circles of radius 2 that can be packed without overlap into a circle of radius 6?

21. What is the total surface area, in cm2, of a regular tetrahedron with edge length 5 cm?

A B C

D

E

F

18. Six points A, B, C, D, E and F are arranged

as shown. How many triangles can be formed with any 3 of the 6 points as vertices?

C A

B E

G H

F