US-TST 2014 Problem 4

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