ILL 1978 Problem 14

Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ 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 $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$ .