RO-TST 2003 Problem 1

Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ $$a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}.$$

Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.