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Problem 1030

AMC 12 late, AIME early

Algebra Difficulty 4.9 Prove it Romanian Mathematical Olympiad · Romania

Find the irrational numbers $x$ with the property that $x^2 + x$ and $x^3 + 2x^2$ are integer numbers.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Denote $x^2 + x = a$ and $x^3 + 2x^2 = b$ . Then $b - a x = x^2 = a - x$ , hence $x(a - 1) = b - a$ . Since $x$ is an irrational number and $a, b$ are integers, we deduce that $a = b = 1$ , and, finally, $x = \frac{-1 \pm \sqrt{5}}{2}$ .

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