Question 9
Let $p_1, p_2, \dots, p_{2009} \in \mathbb{R}^+$. Prove the inequality
$$
\sum_{i=1}^{2009} \frac{p_i}{p_{i+1} + p_{i+2} + \dots + p_{i+1004}} \ge \frac{2009}{1004},
$$
where the indexes are taken mod $2009$.
$$
\sum_{i=1}^{2009} \frac{p_i}{p_{i+1} + p_{i+2} + \dots + p_{i+1004}} \ge \frac{2009}{1004},
$$
where the indexes are taken mod $2009$.