March 2002 Mock Exam

Hungarian Mathematical Olympiad Round 3

A four hour examination

1. The escribed circle to the side AB of ΔABC touches the lines AB, AC (produced) at P, Q respectively. The escribed circle to the side BC touches the lines AC, AB at U, X respectively. Prove that the intersection of P Q and UX is as far from AB as from BC.

2. For which natural numbers n, if any, is there an n-gon containing exactly n^{2} − 30n + 236 acute angles?

3. Let n > 1 be a fixed integer. Find x_{1}, …, x_{n} real numbers so that

∑_{i} x_{i} = 2(n − 1), ∑(x_{i} − 1)^{2} = n

and x_{n} is maximized subject to these conditions.