AUSTRIAN-POLISH 2004 Problem 9

Given are the sequences $$(..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)$$ of positive real numbers. For each integer $n$ the following inequalities hold: $$a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})$$ $$b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})$$ $$c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})$$ Determine $a_{2005}$ , $b_{2005}$ , $c_{2005}$ , if $a_0 = 26, b_0 = 6, c_0 = 2004$ .