Problem: Let $n$ be a positive even integer,
and let $c_1$, $c_2$, \dots, $c_{n-1}$ be real numbers satisfying
\[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \]
Prove that
\[ 2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2 \]
has no real roots.

Solution: We will prove the polynomial is positive for all $x \in \RR$.
As $c_i > 0$, the result is vacuous for $x \le 0$,
so we restrict attention to $x > 0$.

Then letting $c_i = 1 - d_i$ for each $i$,
the inequality we want to prove becomes
\[ x^n + 1 + \frac{x^{n+1}+1}{x+1} > \sum_1^{n-1} d_i x^i
  \qquad\text{given } \sum |d_i| < 1.  \]
But obviously $x^n+1 > x^i$ for any $1 \le i \le n-1$ and $x > 0$.
So in fact $x^n+1 > \sum_1^{n-1} |d_i| x^i$ holds for $x > 0$,
as needed.