olymmath

ISL 2011 A4

    Determine all pairs (f,g)(f, g) of functions from the set of positive integers to itself that satisfy fg(n)+1(n)+gf(n)(n)=f(n+1)−g(n+1)+1f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1 for every positive integer nn. Here, fk(n)f^{k}(n) means f(f(…f⏟k(n)…))\underbrace{f(f(\ldots f}_{k}(n) \ldots)).

    Official text Transcription from NuminaMath olympiads, checked word for word against the official paper. Quoted with source.

    Official text: IMO 2011 Shortlist booklet, page 5.

    Solutions

    Official solution Solution 1 by the IMO Problem Selection Committee

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