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# Pigeonhole Principle

If you have 12 pigeons and there are only 11 roosts, then at least one roost will be quite cozy.

Find the largest integer \(n\) that satisfies the following condition:

If any 6 points are chosen on the perimeter of a circle, then we can draw semicircle of the circle, such that there are at least \(n\) points on it.

Note: A point on the perimeter of the semicircle is considered to lie on it.

Alice and Betty play a game where Alice goes first. They each say a distinct integer from 1 to 15 (inclusive). The first person to say an integer which, when summed with a previously spoken integer, gives the value of 16, will lose the game.

Who will **win** the game?

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