Nemotron-IMO-Bench 2026 A14

Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, $$\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.$$