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Problem 965

AIME late

Algebra Difficulty 5.9 Prove it

Let $n$ be a positive even integer, and let $c_{1}, c_{2}, \ldots, c_{n-1}$ be real numbers satisfying
$\sum_{i=1}^{n-1}\left|c_{i}-1\right|<1 \text {. }$

Prove that
$2 x^{n}-c_{n-1} x^{n-1}+c_{n-2} x^{n-2}-\cdots-c_{1} x^{1}+2$
has no real roots.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

We will prove the polynomial is positive for all $x \in \mathbb{R}$ . As $c_{i}>0$ , the result is vacuous for $x \leq 0$ , so we restrict attention to $x>0$ .
Then letting $c_{i}=1-d_{i}$ for each $i$ , the inequality we want to prove becomes

x^{n}+1+\frac{x^{n+1}+1}{x+1}>\sum_{1}^{n-1} d_{i} x^{i} \quad \text { given } \quad \sum\left|d_{i}\right|x^{i} $for any$ 1 \leq i \leq n-1 $and$ x>0 $. So in fact$ x^{n}+1>\sum_{1}^{n-1}\left|d_{i}\right| x^{i} $holds for$ x>0$, as needed.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.