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an^k, bn^{\ell} , cn^m are sidelengths of a triangle

Source: 2021 Dutch IMO TST 2.4

December 28, 2021

triangle inequalitygeometry

Problem Statement

Determine all positive integers

n

with the following property: for each triple

(a, b, c)

of positive real numbers there is a triple

(k, \ell, m)

of non-negative integer numbers so that

an^k

bn^{\ell}

and

cn^m

are the lengths of the sides of a (non-degenerate) triangle shapes.