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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.

answer type

Expression

is multiple answer

false

language

English

question

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

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

question type

Open-ended

subject

Math

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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: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import