1. Calculate the sum $2\left[h\left(\frac{1}{2009}\right)+h\left(\frac{2}{2009}\right)+\ldots+h\left(\frac{2008}{2009}\right)\right]$, where

$$
h(t)=\frac{5}{5+25^{t}}, \quad t \in R
$$
Solution:

It is observed that the function $h$ is symmetric with respect to the point $\left(\frac{1}{2}, \frac{1}{2}\right)$. Therefore, $h(1-t)=\frac{5}{25^{1-t}+5}=\frac{5 \cdot 5^{t}}{25+5 \cdot 25^{t}}=\frac{25^{t}}{25^{t}+5}$, from which $h(t)+h(1-t)=1$. The sum is then $2 \cdot 1004=2008$.