17th International Mathematical Olympiad 1975 Problems & Solutions

A1.  Let x1 ≥ x2 ≥ ... ≥ xn, and y1 ≥ y2 ≥ ... ≥ yn be real numbers. Prove that if zi is any permutation of the yi, then:       ∑1≤i≤n (xi - yi)2 ≤ ∑1≤i≤n (xi - zi)2. A2.  Let a1 < a2 < a3 < ... be positive integers. Prove that for every i ≥ 1, there are infinitely many an that can be written in the form an = rai + saj, with r, s positive integers and j > i. A3.