MathDB
x^2-rx and x^3-rx are rationals implies x is rational

Source: Czech - Polish - Slovak Match 2013: P3

May 24, 2014
quadraticsalgebrapolynomialalgebra proposed

Problem Statement

For each rational number rr consider the statement: If xx is a real number such that x2−rxx^2-rx and x3−rxx^3-rx are both rational, then xx is also rational.
(a) Prove the claim for r≥43r \ge \frac43 and r≤0r \le 0. (b) Let p,qp,q be different odd primes such that 3p<4q3p <4q. Prove that the claim for r=pqr=\frac{p}q does not hold.
Back to ProblemsView on AoPS