Question 2
A2 (IRE) Let $a_1\ge a_2\ge\cdots\ge a_n$ be real numbers such that
$$a_1^k+a_2^k+\cdots+a_n^k\ge0$$
for all integers $k\gt 0$. Let $p=\max\{|a_1|,\ldots,|a_n|\}$. Prove that $p=a_1$ and that
$$(x-a_1)(x-a_2)\cdots(x-a_n)\le x^n-a_1^n$$
for all $x\gt a_1$.
$$a_1^k+a_2^k+\cdots+a_n^k\ge0$$
for all integers $k\gt 0$. Let $p=\max\{|a_1|,\ldots,|a_n|\}$. Prove that $p=a_1$ and that
$$(x-a_1)(x-a_2)\cdots(x-a_n)\le x^n-a_1^n$$
for all $x\gt a_1$.