KöMaL 2017 B4905

Let $a_1 \ge a_2 \ge a_3 \ge \cdots \ge a_{2n-1} \ge a_{2n} \ge 0$, and $\sum_{i=1}^{2n} a_i = 1$. Prove that $a_1 a_2 + 3 a_3 a_4 + 5 a_5 a_6 + \ldots + (2n-1) a_{2n-1} a_{2n} \le \frac{1}{4}.$ When will the equality hold?