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You and I are going to play a game. We have one million grains of sand in a bag. We take it in turns to remove 2, 3 or 5 grains of sand from the bag. The first person that cannot make a move loses.

Would you go first?

For every natural number k2, find two combinations of k real numbers such that their sum is twice their product.

Show that n2+n+1 is not divisible by 5 for any natural number n.

There are n balls labelled 1 to n. If there are m boxes labelled 1 to m containing the n balls, a legal position is one in which the box containing the ball i has number at most the number on the box containing the ball i+1, for every i.

There are two types of legal moves: 1. Add a new empty box labelled m+1 and pick a box from box 1 to m+1, say the box j. Move the balls in each box with (box) number at least j up by one box. 2. Pick a box j, shift the balls in the boxes with (box) number strictly greater than j down by one box. Then remove the now empty box m.

Prove it is possible to go from an initial position with n boxes with the ball i in the box i to any legal position with m boxes within n+m legal moves.

We can define the absolute value |x| of any real number x as follows. |x|=x if x0 and |x|=x if x<0. What are |3|, |4.3| and |0|?

Prove that |x|0.

Prove that |x|x. It may be helpful to compare each of |3|, |4.3| and |0| with 3, 4.3 and 0 respectively.

Two fractions sum up to 1, but their difference is 110. What are they?

On her birthday, my grandma was asked how old she was. She said: "Start with the year I was born. Add the current year to it. Then, from the sum subtract the year I celebrated by 20th birthday. From that, take away the year I was 30. The result will be 16." How old is my grandma?

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In the long addition above, each letter corresponds to a different digit. What is the sum D+O+G+C+A+T?

Let ABCDE be a regular pentagon. The point G is the midpoint of CD, the point F is the midpoint of AE. The lines EG and BF intersect at the point H. Find the angle EHF.

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