ELMO 2014 Problem 4

Let $n$ be a positive integer and let $a_1$ , $a_2$ , \dots, $a_n$ be real numbers strictly between $0$ and $1$ . For any subset $S$ of $\{1, 2, ..., n\}$ , define $$f(S) = \prod_{i \in S} a_i \cdot \prod_{j \not \in S} (1-a_j).$$ Suppose that $\sum_{|S| \text{ odd}} f(S) = \frac{1}{2}$ . Prove that $a_k = \frac{1}{2}$ for some $k$ . (Here the sum ranges over all subsets of $\{1, 2, ..., n\}$ with an odd number of elements.)

Proposed by Kevin Sun