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Two polynomials cannot be perfect cubes at the same time

Source: 2022 Junior Macedonian Mathematical Olympiad P5

June 7, 2022
polynomialnumber theory

Problem Statement

Let nn be a positive integer such that n5+n3+2n2+2n+2n^5+n^3+2n^2+2n+2 is a perfect cube. Prove that 2n2+n+22n^2+n+2 is not a perfect cube.
Proposed by Anastasija Trajanova
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