Question

Suppose x is a nonzero real number such that both x⁵ and 20x + 19/x are rational numbers. Prove that x is a rational number.

Answer 1

Since x⁵ is rational, we see that (20x)⁵ and (x/19)⁵ are rational numbers. But

is also rational. Thus T is a rational number and T ≠ 0. We conclude that 20x −(19/x) is a rational number. This combined with the given condition that 20x + (19/x) is rational shows 2 · 20 · x is rational. Therefore x is rational.