RO-NMO 2010 Problem 101

Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that $$\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.$$ Prove that $(a_n)_{n\ge0}$ is a geometric sequence.

Lucian Dragomir