Problem 1

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size.

Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.

Problem 2

Let a1 , a2 , . . . , an be distinct positive integers and let M be a set of n− 1 positive integers not containing s = a1 + a2 +· · · + an . A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a1 , a2 , . . . , an in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M .

Problem 3

Consider five points A,B,C,D and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let "l" be a line passing through A. Suppose that l intersects the interior of the segment DC at F and intersects line BC at G. Suppose also that EF=EC=EG. Prove that l is the bisector of angle DAB.

P.S. - Was racking my brains all night long and couldn't even understand problem 1!! Question 3 was easy to understand but is quite tough to prove unless you come in the reverse order from considering the line l to be a bisector... couldn't complete it though... will post the answers when I find them out or get hold of of them on the net.