IMO 1982 SL 19

Problem

C7 (CZS 3) Let M be the set of real numbers of the form m+n \sqrt m2+n2 , where m and n are positive integers. Prove that for every pair x \inM, y \inM with x < y, there exists an element z \inM such that x < z < y.

Solution

Let us set x = m/n. Since f(x) = (m + n)/ \sqrt m2 + n2 = (x + 1)/ \sqrt 1 + x2 is a continuous function of x, f(x) takes all values between any two values of f; moreover, the corresponding x can be rational. This completes the proof. Remark. Since f is increasing for x \geq1, 1 \leqx < z < y implies f(x) < f(z) < f(y).