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IMO Shortlist 2012, Number Theory 5

Source: IMO Shortlist 2012, Number Theory 5

July 29, 2013

algebranumber theoryIMO Shortlistprime numberspolynomial

Problem Statement

For a nonnegative integer

n

define

\operatorname{rad}(n)=1

if

n=0

or

n=1

, and

\operatorname{rad}(n)=p_1p_2\cdots p_k

where

p_1<p_2<\cdots <p_k

are all prime factors of

n

. Find all polynomials

f(x)

with nonnegative integer coefficients such that

\operatorname{rad}(f(n))

divides

\operatorname{rad}(f(n^{\operatorname{rad}(n)}))

for every nonnegative integer

n

.

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