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Omni-MATH / Let mathbb R be the set of real numbers. We denote by mathcal F the set of all functions fcolonmathbb Rtomathbb R such that f(x+f(y))=f(x)+f(y) for every x,yinmathbb R Find all rat…

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problem

Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that f(x+f(y))=f(x)+f(y)f(x + f(y)) = f(x) + f(y) for every $x,y\in\mathbb R$ Find all rational numbers qq such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying f(z)=qzf(z)=qz.
Plain-text mathematical notation (without MathML)
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
f(x+f(y))=f(x)+f(y)
for every $x,y\in\mathbb R$ Find all rational numbers q such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying f(z)=qz.
Original LaTeX notation
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
$$f(x + f(y)) = f(x) + f(y)$$
for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

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Source and history

task_id
26877e62-d332-5747-a3a4-f593321ea45f
task_key
test--26877e62-d332-5747-a3a4-f593321ea45f
task_revision_id
1
config / split
/ test

Official source

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