OIMU 2010 Problem 4

Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$ , with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$ , at least one of the coefficients $a_k,a_{k+1}$ is different from zero.