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OlympiadBench / 1998 / Let R_(>0) be the set of positive real numbers. Find all functions…

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

Let

\mathbb{R}_{>0}

be the set of positive real numbers. Find all functions

f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}

such that, for every

x \in \mathbb{R}_{>0}

, there exists a unique

y \in \mathbb{R}_{>0}

satisfying

x f(y)+y f(x) \leqslant 2 .

Plain-text mathematical notation (without MathML)

Let R_(>0) be the set of positive real numbers. Find all functions f:R_(>0)→R_(>0) such that, for every x∈R_(>0), there exists a unique y∈R_(>0) satisfying

xf(y)+yf(x)≤2.

Original LaTeX notation

Let $\mathbb{R}_{>0}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that, for every $x \in \mathbb{R}_{>0}$, there exists a unique $y \in \mathbb{R}_{>0}$ satisfying

$$
x f(y)+y f(x) \leqslant 2 .
$$

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1998
task_id: 8524922d-ab1e-505a-a6d3-b10b25fa0ce2
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1998
task_revision_id: 2
config / split: OE_TO_maths_en_COMP / train

Official source

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