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 one can construct a graph with no triangles and whose chromatic number is at least .
Source: T. Andreescu & G. Dospinescu, Problems from the Book, pp. 90 & 130.