Nemotron-IMO-Bench 2026 A36

Let $n>1$ be an integer. Find the largest positive integer $k$ such that for all real numbers $a_1\geq a_2\geq\ldots\geq a_n\geq 0$ $$\left(\frac{a_1+\ldots+a_k}{k}\right)^2\geq \frac{a_1^2+\ldots+a_n^2}{n}.$$