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Omni-MATH / Let Q be the set of rational numbers. A function f:Q→Q is called aquaesulian if the following property holds: for every x,y∈Q, f(x+f(y))=f(x)+y or f(f(x)+y)=x+f(y). Show that there…

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Let Q\mathbb{Q} be the set of rational numbers. A function f:Q→Qf: \mathbb{Q} \to \mathbb{Q} is called aquaesulian if the following property holds: for every x,y∈Qx,y \in \mathbb{Q}, f(x+f(y))=f(x)+yorf(f(x)+y)=x+f(y). f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). Show that there exists an integer cc such that for any aquaesulian function ff there are at most cc different rational numbers of the form f(r)+f(−r)f(r) + f(-r) for some rational number rr, and find the smallest possible value of cc.
Plain-text mathematical notation (without MathML)
Let Q be the set of rational numbers. A function f:Q→Q is called aquaesulian if the following property holds: for every x,y∈Q,
f(x+f(y))=f(x)+y or f(f(x)+y)=x+f(y).
Show that there exists an integer c such that for any aquaesulian function f there are at most c different rational numbers of the form f(r)+f(−r) for some rational number r, and find the smallest possible value of c.
Original LaTeX notation
Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$,
\[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \]
Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$, and find the smallest possible value of $c$.

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task_id
d8afe04e-5f3e-5764-bb01-9d84b35ea862
task_key
test--d8afe04e-5f3e-5764-bb01-9d84b35ea862
task_revision_id
1
config / split
/ test

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