f(2)=4/5
f(1/2)=(1/4)/(5/4)=1/5
f(3)=9/10
f(1/3)=(1/9)/(10/9)=1/10
f(x)+f(1/x)=x²/(1+x²) +(1/x)²/(1 +1/x²) =x²/(1+x²) +1/(x²+1) =(x²+1)/(x²+1)=1
所以 f(1)+f(2)+...+f(2012)+f(1/2)+f(1/3)+...+f(1/2012)
=f(1)+[f(2)+f(1/2)]+[f(3)+f(1/3)]+...+[f(2012)+f(1/2012)]
=f(1)+1+1+1+...+1
=1/2 +2011
=4023/2