IMAS 2012–2013 — Junior Division — Round 2 Question 2

Into the expression $y=\frac{x^2}{1+x^2}$, we substitute for $x$ the numbers $\frac1{2012},\frac1{2008},\frac1{2004},\ldots,\frac14,4,8,12,\ldots,2012$. What is the sum of all the $y$ values?