Suppose that $x$ and $y$ are positive real numbers such that $x^3, y^3$ and $x + y$ are all rational numbers. Show that the numbers $xy, x^2+y^2, x$ and $y$ are also rational
Source: Norwegian Mathematical Olympiad 2001 - Abel Competition p1b
Tags: rational, algebra
Suppose that $x$ and $y$ are positive real numbers such that $x^3, y^3$ and $x + y$ are all rational numbers. Show that the numbers $xy, x^2+y^2, x$ and $y$ are also rational