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\lfloor a^{2014} \rfloor + \lfloor b^{2014} \rfloor =\lfloor a \rfloor ^{2014}

Source: INAMO Shortlist 2014 A1

May 21, 2019

Flooralgebrafloor function

Problem Statement

Let

a, b

be positive real numbers such that there exist infinite number of natural numbers

k

such that

\lfloor a^k \rfloor + \lfloor b^k \rfloor = \lfloor a \rfloor ^k + \lfloor b \rfloor ^k

. Prove that

\lfloor a^{2014} \rfloor + \lfloor b^{2014} \rfloor = \lfloor a \rfloor ^{2014} + \lfloor b \rfloor ^{2014}