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OlympiadBench / 1687 / Determine all pairs (f,g) of functions from the set of positive integers to…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

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)+1 f^{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 $\underbrace{f(f(\ldots f}_{k}(n) \ldots))$.
Plain-text mathematical notation (without MathML)
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 $\underbrace{f(f(\ldots f}_{k}(n) \ldots))$.
Original LaTeX notation
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 $\underbrace{f(f(\ldots f}_{k}(n) \ldots))$.

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

Discussion

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Source and history

upstream_id / Original problem ID
1687
task_id
240ca69e-8fc5-551a-8166-224534bb617e
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1687
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import