CWMO 2012 Problem 6

Define a sequence $\{a_n\}$ by$$a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),$$ find integer $k$ such that $a_{k}<1<a_{k+1}.$ (September 29, 2012, Hohhot)