# 2021-19 The answer is zero

Suppose that $$a_1 + a_2 + \dots + a_n =0$$ for real numbers $$a_1, a_2, \dots, a_n$$ and $$n \geq 2$$. Set $$a_{n+i}=a_i$$ for $$i=1, 2, \dots$$. Prove that
$\sum_{i=1}^n \frac{1}{a_i (a_i+a_{i+1}) (a_i+a_{i+1}+a_{i+2}) \dots (a_i+a_{i+1}+\dots+a_{i+n-2})} =0$
if the denominators are nonzero.

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