Baltic Way 2021 · Problem 2

We claim that

$$a^2 + bc > \frac{a^2 + b^2 + c^2}{2},$$

which will finish the proof. Note that the claimed inequality is equivalent to

$$\begin{aligned} a^2 + bc > \frac{a^2 + b^2 + c^2}{2} &\iff 2a^2 + 2bc > a^2 + b^2 + c^2 \\ &\iff a^2 > (b-c)^2 \iff a > |b-c|, \end{aligned}$$

which holds due to the assumption of $a, b, c$ being side lengths of a triangle.