2006 IMO Shortlist A2

The sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ is defined recursively by $$a_{0}=-1, \quad \sum_{k=0}^{n} \frac{a_{n-k}}{k+1}=0 \quad \text { for } \quad n \geq 1$$ Show that $a_{n}>0$ for $n \geq 1$. (Poland)

实数序列 $a_{0}, a_{1}, a_{2}, \ldots$ 递归地定义为 $$a_{0}=-1, \quad \sum_{k=0}^{n} \frac{a_{n-k}}{k+1}=0 \quad \text { for } \quad n \geq 1$$ 证明 $a_{n}>0$ for $n \geq 1 美元。 (波兰)