MathDB

IMO's Hojoo Lee vs. USAMO's Harazi

Source: USAMO 2006, Problem 3, proposed by Titu Andreescu and Gabriel Dospinescu

April 20, 2006

algebrapolynomialmodular arithmetic

Problem Statement

For integral

m

, let

p(m)

be the greatest prime divisor of

m.

By convention, we set

p(\pm 1) = 1

and

p(0) = \infty.

Find all polynomials

f

with integer coefficients such that the sequence

\{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0}

is bounded above. (In particular, this requires

f \left (n^2 \right ) \neq 0

for

n \geq 0.

)

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