Question 130
Assume that positive real numbers $a_1, a_2, \dots, a_{2006}$ satisfy the equality
$$
\frac{a_1}{(a_1+a_2)a_2} + \frac{a_2}{(a_2+a_3)a_3} + \dots + \frac{a_{2005}}{(a_{2005}+a_{2006})a_{2006}} + \frac{a_{2006}}{(a_{2006}+a_1)a_1} = 2006
$$
Find the value of the expression
$$
\frac{a_1}{a_{2006}(a_{2006} + a_1)} + \frac{a_2}{a_1(a_1 + a_2)} + \dots + \frac{a_{2005}}{a_{2004}(a_{2004} + a_{2005})} + \frac{a_{2006}}{a_{2005}(a_{2005} + a_{2006})}
$$
$$
\frac{a_1}{(a_1+a_2)a_2} + \frac{a_2}{(a_2+a_3)a_3} + \dots + \frac{a_{2005}}{(a_{2005}+a_{2006})a_{2006}} + \frac{a_{2006}}{(a_{2006}+a_1)a_1} = 2006
$$
Find the value of the expression
$$
\frac{a_1}{a_{2006}(a_{2006} + a_1)} + \frac{a_2}{a_1(a_1 + a_2)} + \dots + \frac{a_{2005}}{a_{2004}(a_{2004} + a_{2005})} + \frac{a_{2006}}{a_{2005}(a_{2005} + a_{2006})}
$$