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IMO Shortlist 2011, Algebra 4

Source: IMO Shortlist 2011, Algebra 4

July 11, 2012

functionalgebrafunctional equationIMO Shortlist

Problem Statement

Determine all pairs

(f,g)

of functions from the set of positive integers to itself that satisfy

f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1

for every positive integer

n

. Here,

f^k(n)

means

\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))

.
Proposed by Bojan Bašić, Serbia