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Convolution of order f(n)

Source: 2019 USAMO Problem 1

April 17, 2019
function2019 Usamo problem 1Hi

Problem Statement

Let N\mathbb{N} be the set of positive integers. A function f:N→Nf:\mathbb{N}\to\mathbb{N} satisfies the equation f(f(…f⏟f(n) times(n)…))=n2f(f(n))\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))} for all positive integers nn. Given this information, determine all possible values of f(1000)f(1000).
Proposed by Evan Chen
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