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

$$\pm a_n x^n \pm a_{n-1}x^{n-1} \pm \dots \pm a_1 x \pm a_0$$

has $n$ distinct real roots.