1996 IMO 第 2 题

A2 (IRE) Let $a_{1} \geq a_{2} \geq \cdots \geq a_{n}$ be real numbers such that $$a_{1}^{k}+a_{2}^{k}+\cdots+a_{n}^{k} \geq 0$$ for all integers $k>0$. Let $p=\max \left\{\left|a_{1}\right|, \ldots,\left|a_{n}\right|\right\}$. Prove that $p=a_{1}$ and that $$\left(x-a_{1}\right)\left(x-a_{2}\right) \cdots\left(x-a_{n}\right) \leq x^{n}-a_{1}^{n}$$ for all $x>a_{1}$.

A2 (IRE) 设 $a_{1} \geq a_{2} \geq \cdots \geq a_{n}$ 为实数，使得 $$a_{1}^{k}+a_{2}^{k}+\cdots+a_{n}^{k} \geq 0$$ 对于所有整数 $k>0$ 为实数。令 $p=\max \left\{\left|a_{1}\right|, \ldots,\left|a_{n}\right|\right\}$。证明 $p=a_{1}$ 和 $$\left(x-a_{1}\right)\left(x-a_{2}\right) \cdots\left(x-a_{n}\right) \leq x^{n}-a_{1}^{n}$$ 对于所有 $x>a_{1}$。