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Functional Inequality on N

Source: HMMT Invitational Contest 2016, Problem 3

April 22, 2016
HMMTalgebraFunctional inequalityfunctional equationHMIC

Problem Statement

Denote by N\mathbb{N} the positive integers. Let f:N→Nf:\mathbb{N} \rightarrow \mathbb{N} be a function such that, for any w,x,y,z∈Nw,x,y,z \in \mathbb{N}, f(f(f(z)))f(wxf(yf(z)))=z2f(xf(y))f(w). f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). Show that f(n!)≥n!f(n!) \ge n! for every positive integer nn.
Pakawut Jiradilok
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