Prove that if $a \leqslant c$, then $a^{2}+b^{2} \leqslant 5 c^{2}$.

## Solution

Assume that $c$ is not the smallest side, for example, $a \leqslant c$. Then

$$a^{2} \leqslant c^{2}, b^{2}<(a+c)^{2} \leqslant(2 c)^{2}=4 c^{2}$$

Therefore, $a^{2}+b^{2} \leqslant 5 c^{2}$, which contradicts the condition.