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MEMO 2010, Problem T-7: Decomposition into cubes and squares

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September 12, 2010
geometry3D geometrynumber theory proposednumber theory

Problem Statement

For a nonnegative integer nn, define ana_n to be the positive integer with decimal representation 1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.} Prove that an3\frac{a_n}{3} is always the sum of two positive perfect cubes but never the sum of two perfect squares.
(4th Middle European Mathematical Olympiad, Team Competition, Problem 7)
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