CAPS 2013 Problem 3

For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2-rx$ and $x^3-rx$ are both rational, then $x$ is also rational.

list Prove the claim for $r \ge \frac43$ and $r \le 0$ . (b) Let $p,q$ be different odd primes such that $3p <4q$ . Prove that the claim for $r=\frac{p}q$ does not hold. [/list]