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Prove that one can find...

Source: Indian IMOTC 2005 Day 1 Problem 2

September 17, 2005
geometry3D geometrynumber theory solvednumber theory

Problem Statement

Prove that one can find a n0∈Nn_{0} \in \mathbb{N} such that ∀m≥n0\forall m \geq n_{0}, there exist three positive integers aa, bb , cc such that (i) m3<a<b<c<(m+1)3m^3 < a < b < c < (m+1)^3; (ii) abcabc is the cube of an integer.
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