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2024 IMO Problems/Problem 6

Revision as of 12:26, 30 July 2024 by Codemaster11 (talk | contribs)

Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called $\emph{aquaesulian}$ 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 $c$.

Video Solution

https://youtu.be/7h3gJfWnDoc

See Also

2024 IMO (Problems) • Resources
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