IMC 2014 Problem 3

Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial $$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$ has $n$ distinct real roots.

(Proposed by Stephan Neupert, TUM, München)