1996 IMO 第 4 题

A4 (KOR) Let $a_{1}, a_{2}, \ldots, a_{n}$ be nonnegative real numbers, not all zero. (a) Prove that $x^{n}-a_{1} x^{n-1}-\cdots-a_{n-1} x-a_{n}=0$ has precisely one positive real root. (b) Let $A=\sum_{j=1}^{n} a_{j}, B=\sum_{j=1}^{n} j a_{j}$, and let $R$ be the positive real root of the equation in part (a). Prove that $$A^{A} \leq R^{B}$$

A4 (KOR) 令 $a_{1}, a_{2}, \ldots, a_{n}$ 为非负实数，且不全为零。 (a) 证明 $x^{n}-a_{1} x^{n-1}-\cdots-a_{n-1} x-a_{n}=0$ 恰好有一个正实根。 (b) 令$A=\sum_{j=1}^{n} a_{j}, B=\sum_{j=1}^{n} j a_{j}$，并令$R$为(a)部分方程的正实根。证明 $$A^{A} \leq R^{B}$$