2. in the Diagram Shown, You Are Given

Question Set #1 Team Number:

n  3 1. Given that n is an integer, find the values of n for which is an integer. n  1

A 2. In the diagram shown, you are given: length of AB = 2 units, length of BC = 1 unit, 2 angle ABC = 135º, angle ACB = xº, where 10  4 2 sin x°  ,and k is an integer. B 135º k

Find the value of k. 1 xº

3. Three fair (regular, six-sided) dice are rolled, and their scores are added together. What is the probability that the sum of the scores is 7?

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Answer Question 1 Question 2 Question 3 (a)

ISMTFJunior/Paris/2010/Team/Set#1 Question Set #2 Team Number:

4. Find 1002  992  982  97 2  962  952  . . . . .  22  11

5. A basin is filled with water from two taps. On its own, one of the taps could fill the basin in 40 seconds, whilst the other would take one minute. How long would it take to fill the basin using both taps at the same time, giving your answer in seconds?

C 6. Triangle ABC is constructed with four isosceles triangles. Find x so that PQ is parallel to CB.

xº A Q B

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1 7. When calculating the decimal value of the fraction , the answer begins: 81 1  0.01234 . . . . State the sum of the digits from the first to the tenth decimal place. 81

8. Using only letters from the set {B, C, D, F, G, H, J}, all possible four letter words are written in alphabetical order starting with BBBB. What is the 2010th word?

9. The diagram shows a regular decagon (10-sided polygon). Find the sum x°  y°  z°

x y

K z

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Question Set #4 Team Number: ISMTFJunior/Paris/2010/Team/Set#3 10. The faces of a cuboid have areas 48 cm 2 , 54 cm 2 , and 72 cm 2 . Find its volume.

2  1  11. If x2  3x  1  0 , then the value of x2    is a positive integer. Find this integer.  x 

12. In the diagram, ST is a tangent to the circle ABCD at A, and DB is a diameter of the circle. Find angle CBD C

D B 34º

38º S A T

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Question Set #5 Team Number:

ISMTFJunior/Paris/2010/Team/Set#4 13. The price of a ticket is decreased by 20%, but it remains unsold and the reduced price is then subject to a further 20% decrease. By exactly what percentage must the twice reduced price be increased in order to restore it to its original value?

2 1 10   10 2 1 p 2378 14. If A    , and you are given that A       , find p and q. 1 0 1 0 q 985 

15. A cylinder with a base radius of R cm and length 3R cm is made hollow by drilling a hole of 1 radius R cm throughout its length, as shown. 3 The surface area of the hollow cylinder can be a written as  R 2 cm 2 . Find a. 9

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Question Set #6 Team Number: ISMTFJunior/Paris/2010/Team/Set#5 16. If there are 20 people in a room, and they all shake hands with each other, apart from John, James, Jeremy, Jim, and Jeff, who do not shake hands with each other, how many handshakes are there altogether?

17. There are 15 beads in a bag, with 7 blue, 5 red, and 3 green beads. A bead is selected at random, and not replaced. Then a second bead is selected without replacement, and then a third is selected without replacement. The probability that the beads k are not all of the same colour can be written in its simplest form as , where k is an 455 integer. Find k

q 18. In solving the system of equations below, it is found that y  p  a , where p, q, and r are r positive integers. Find p, q, and r.

3x  2y  z  a    2x  y  3z  13    x  2y  3z  4

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19. The number 23  34  45  56 is written out in full. How many zeros are there at the end of the number?

20 Find the positive real solution of the equation 14x4  121x2  675

21. A square is drawn inside a semicircle, as shown. The ratio of the area of the semicircle to the a square can be written as   , where b a and b are integers. Find a and b

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ISMTFJunior/Paris/2010/Team/Set#7