2021 IMO Shortlist A5

Let $n \geq 2$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots+a_{n}=1$. Prove that $$\sum_{k=1}^{n} \frac{a_{k}}{1-a_{k}}\left(a_{1}+a_{2}+\cdots+a_{k-1}\right)^{2}<\frac{1}{3}$$

令 $n \geq 2$ 为整数，并令 $a_{1}, a_{2}, \ldots, a_{n}$ 为正实数，使得 $a_{1}+a_{2}+\cdots+a_{n}=1$。证明 $$\sum_{k=1}^{n} \frac{a_{k}}{1-a_{k}}\left(a_{1}+a_{2}+\cdots+a_{k-1}\right)^{2}<\frac{1}{3}$$