BG-MO 1987 Problem 1

Let $f(x)=x^n+a_1x^{n-1}+\ldots+a_n~(n\ge3)$ be a polynomial with real coefficients and $n$ real roots, such that $\frac{a_{n-1}}{a_n}>n+1$ . Prove that if $a_{n-2}=0$ , then at least one root of $f(x)$ lies in the open interval $\left(-\frac12,\frac1{n+1}\right)$ .