Given natural numbers $m, n$ and $X$ such that $X \ge m$ and $X \ge n$. Prove that one can find two integers $u$ and $v$ such that $|u| + |v| > 0$, $|u| \le \sqrt{X}$, $|v| \le \sqrt{X}$ and $$0 \le mu + nv \le 2 \sqrt{X}.$$
Source: 2016 Latvia BW TST P2
Tags: number theory, inequalities, algebra
Given natural numbers $m, n$ and $X$ such that $X \ge m$ and $X \ge n$. Prove that one can find two integers $u$ and $v$ such that $|u| + |v| > 0$, $|u| \le \sqrt{X}$, $|v| \le \sqrt{X}$ and $$0 \le mu + nv \le 2 \sqrt{X}.$$