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Berkeley Math Circle: Monthly Contest 5

Algebra Difficulty 4.7 AIME Prove it United States

Problem:

Prove that if $x$ is a positive real number such that $x + x^{-1}$ is an integer, then $x^{3} + x^{-3}$ is an integer as well.

Solution

Solution:

This follows from the identity
$x^{3} + \frac{1}{x^{3}} = \left(x + \frac{1}{x}\right)^{3} - 3\left(x + \frac{1}{x}\right).$

If $x + x^{-1}$ is an integer, then so is $\left(x + x^{-1}\right)^{3} - 3\left(x + x^{-1}\right)$ , and thus $x^{3} + x^{-3}$ is an integer.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.