Problem

Source: 2013 Romania NMO VIII p4

Tags: rational, algebra



A set $M$ of real numbers will be called special if it has the properties: (i) for each $x, y \in M, x\ne y$, the numbers $x + y$ and $xy$ are not zero and exactly one of them is rational; (ii) for each $x \in M, x^2$ is irrational. Find the maximum number of elements of a special set.