Math Homework Problems 2

Homeworks for to IMO 2010 Training Camp (Senior). This homework is due on 22 January 2010.

1. Ivan and Peter alternatively write down 0 or 1 until each of them has written 2001 digits. Peter is a winner if the number, whose binary representations has been obtained, cannot be expressed as the sum of two perfect squares. Prove that Peter has a winning strategy whenever Ivan starts.

2. Prove that for every n one can construct a graph with no triangles and whose chromatic number is at least n.

Source: T. Andreescu & G. Dospinescu, Problems from the Book, pp. 90 & 130.

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