SMO 2010 Open Round 1 Question 13

Let $a_1=1$, $a_2=2$, and $a_{n+1}=\dfrac{2n}{n+1}a_n-\dfrac{n-1}{n+1}a_{n-1}$ for $n\ge2$. It is known that $a_n\gt 2+2009/2010$ for every $n\ge m$, where $m$ is a positive integer. Find the least $m$.