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Integers satisfying squared and product sum

Source: IMO Shortlist 1995, A2

August 10, 2008
algebrasystem of equationsequationsIMO Shortlist

Problem Statement

Let a a and b b be non-negative integers such that ab≥c2, ab \geq c^2, where c c is an integer. Prove that there is a number n n and integers x1,x2,…,xn,y1,y2,…,yn x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n such that \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.
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