parmenides51 wrote: Fix $a_1, . . . , a_n \in (0, 1)$ and define $$f(I) = \prod_{i \in I} a_i \cdot \prod_{j \notin I} (1 - a_j)$$for each $I \subseteq \{1, . . . , n\}$. Assuming that $$\sum_{I\subseteq \{1,...,n\}, |I| odd} {f(I)} = \frac12,$$show that at least one $a_i$ has to be equal to $\frac12$. (Paolo Leonetti) Let $I_n= \{1,...,n\}$, define $b_i=1-a_i$ for all $i \in I_n$. Consider the polynomial $P(X)=\prod_{i=1}^{n}(a_iX-b_i)$. We want to prove $P(1)=0$. Note that the condition is equivalent to $P(-1)-P(1)=(-1)^n$, but $P(-1)=(-1)^n$ (since $a_i+b_i=1$), hence the result