1. ## Sum of functions

Let $f(x)=\frac{x^{2}} {1+x^2}$

Compute:

$f(\frac{1}{1}) + f(\frac{1}{2})+...+f(\frac{1}{10}) + f(\frac{2}{1})+f(\frac{2}{2})+...+f(\frac{2}{10})+ f(\frac{3}{1})+f(\frac{3}{2})+...+f(\frac{3}{10})+ ...+f(\frac{10}{1})+f(\frac{10}{2})+...+f(\frac{10 }{10})$

2. Notice that $f(\frac{1}{1})=f(\frac{2}{2})=f(\frac{3}{3})=...=f (\frac{10}{10})=f(1)$
So 10 out of your 100 terms give f(1)
$f(\frac{1}{2})=f(\frac{2}{4})=f(\frac{3}{6})=f(\fr ac{4}{8})=f(\frac{5}{10})$
and
$f(\frac{2}{1})=f(\frac{4}{2})=f(\frac{6}{3})=f(\fr ac{8}{4})=f(\frac{10}{5})$
5 of these terms give f(0.5) and 5 more give f(2).

Already, we have regrouped 20 out of 100 of these terms in 3 simple operations: 10f(1)+5f(0.5)+5f(2).

I suspect the 80 remaining could also be regrouped like this!

3. Notice that $f\left( x \right) + f\left( {\tfrac{1}{x}} \right) = 1$