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[f^2(m)+f(n)]|(m^2+n)^2

Source: IMO ShortList 2004, number theory problem 3

February 19, 2005
functionnumber theoryalgebraDivisibilityfunctional equationIMO Shortlist

Problem Statement

Find all functions f:N∗→N∗ f: \mathbb{N^{*}}\to \mathbb{N^{*}} satisfying (f2(m)+f(n))∣(m2+n)2 \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2} for any two positive integers m m and n n.
Remark. The abbreviation N∗ \mathbb{N^{*}} stands for the set of all positive integers: N∗={1,2,3,...} \mathbb{N^{*}}=\left\{1,2,3,...\right\}. By f2(m) f^{2}\left(m\right), we mean (f(m))2 \left(f\left(m\right)\right)^{2} (and not f(f(m)) f\left(f\left(m\right)\right)).
Proposed by Mohsen Jamali, Iran
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