HMMT February, 2024

Number theory Difficulty 5.0 AIME, harder Prove it United States

Problem:

Determine, with proof, whether there exist positive integers $x$ and $y$ such that $x+y$ , $x^{2}+y^{2}$ , and $x^{3}+y^{3}$ are all perfect squares.

Solution

Solution:

Take $(x, y) = (184, 345)$ . Then $x+y = 23^{2}$ , $x^{2}+y^{2} = 391^{2}$ , and $x^{3}+y^{3} = 6877^{2}$ .

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