Question 6
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 $c$.
$$f(x+f(y))=f(x)+y\quad\text{or}\quad f(f(x)+y)=x+f(y).$$
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 $c$.