IMO 1978 LL FIN14

Origin: FIN

Problem

Let p(x, y) and q(x, y) be polynomials in two variables such that for x \geq0, y \geq0 the following conditions hold: (i) p(x, y) and q(x, y) are increasing functions of x for every fixed y. (ii) p(x, y) is an increasing and q(x) is a decreasing function of y for every fixed x. (iii) p(x, 0) = q(x, 0) for every x and p(0, 0) = 0. Show that the simultaneous equations p(x, y) = a, q(x, y) = b have a unique solution in the set x \geq0, y \geq0 for all a, b satisfying 0 \leqb \leqa but lack a solution in the same set if a < b.