2015 IMO Shortlist A1

Suppose that a sequence $a_{1}, a_{2}, \ldots$ of positive real numbers satisfies $$a_{k+1} \geq \frac{k a_{k}}{a_{k}^{2}+(k-1)}$$ for every positive integer $k$. Prove that $a_{1}+a_{2}+\cdots+a_{n} \geq n$ for every $n \geq 2$.

假设正实数序列 $a_{1}, a_{2}, \ldots$ 对于每个正整数 $k$ 都满足 $$a_{k+1} \geq \frac{k a_{k}}{a_{k}^{2}+(k-1)}$$。证明 $a_{1}+a_{2}+\cdots+a_{n} \geq n$ 对于每个 $n \geq 2$。