Question 4
A4 (KOR) Let $a_1,a_2,\ldots,a_n$ be nonnegative real numbers, not all zero.
(a) Prove that $x^n-a_1x^{n-1}-\cdots-a_{n-1}x-a_n=0$ has precisely one positive real root.
(b) Let $A=\sum_{j=1}^na_j$, $B=\sum_{j=1}^nja_j$, and let $R$ be the positive real root of the equation in part (a). Prove that
$$A^A\le R^B.$$
(a) Prove that $x^n-a_1x^{n-1}-\cdots-a_{n-1}x-a_n=0$ has precisely one positive real root.
(b) Let $A=\sum_{j=1}^na_j$, $B=\sum_{j=1}^nja_j$, and let $R$ be the positive real root of the equation in part (a). Prove that
$$A^A\le R^B.$$