What is the value of (2019 − (2000 − (10−9))) − (2000 − (10 − (9 − 2019))) (2019 − (2000 − (10−9))) − (2000 − (10 − (9 − 2019))) ?

3. Used in measuring the width of a wire, one mil is equal to one thousandth of an inch. An inch is about 2.5 cm.

Which of these is approximately equal to one mil?

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4. For how many positive integer values of n is n² 2+2n prime?

6. Each of the factors of 100 is to be placed in a 3 by 3 grid, one per cell, in such a way that the products of the three numbers in each row, column and diagonal are all equal. The positions of the numbers 1, 2, 50 and x are shown in the diagram.

What is the value of x?

9. A square of paper is folded in half four times to obtain a smaller square. Then a corner is removed as shown.

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Which of the following could be the paper after it is unfolded?

10. Which of the following five values of n is a counter example to the statement in the box below?

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11.

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12. A circle with radius 1 touches the sides of a rhombus, as shown. Each of the smaller angles between the sides of the rhombus is 60∘.

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What is the area of the rhombus?

13. Anish has a number of small congruent square tiles to use in a mosaic. When he forms the tiles into a square of side n, he has 64 tiles left over. When he tries to form the tiles into a square of side n + 1, he has 25 too few.

How many tiles does Anish have?

14. One of the following is the largest square that is a factor of 10!. Which one?

Note that, n! = 1 × 2 × 3 × ∙∙∙× ( n − 1 ) × n.

15. The highest common factors of all the pairs chosen from the positive integers Q, R and S are three different primes.

What is the smallest possible value of Q + R + S?