Nemotron-IMO-Bench 2026 C26

The positive integers $a,\ b,\ c$ are pairwise relatively prime. Let $g(a, b, c)$ be the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that $$g(a, b, c)\ge \sqrt{2abc}.$$