Problem

Source: 2009 Romania JBMO TST 1.3

Tags: number theory, Sets, Product



Let $A$ be a finite set of positive real numbers satisfying the property: For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity. Show that the product of all elements in $A$ is equal to $1$.