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APMO 2017: Powerful rationals

Source: APMO 2017, problem 4

May 14, 2017

number theoryAPMOLTE LemmaLifting the Exponent

Problem Statement

Call a rational number

r

powerful if

r

can be expressed in the form

\dfrac{p^k}{q}

for some relatively prime positive integers

p, q

and some integer

k >1

. Let

a, b, c

be positive rational numbers such that

abc = 1

. Suppose there exist positive integers

x, y, z

such that

a^x + b^y + c^z

is an integer. Prove that

a, b, c

are all powerful.
Jeck Lim, Singapore

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