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functions from {1,2, ..., n} to {1,2, ..., n}

Source: RMO District 2005, 10th Grade, Problem 4

March 5, 2005
functioninductionalgebrafunctional equationalgebra proposed

Problem Statement

Let n≥3n\geq 3 be an integer. Find the number of functions f:{1,2,…,n}→{1,2,…,n}f:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\} such that f(f(k))=f3(k)−6f2(k)+12f(k)−6,  for all k≥1. f(f(k)) = f^3(k) - 6f^2(k) + 12f(k) - 6 , \ \textrm{ for all } k \geq 1 .
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