2024 IMO Problems/Problem 6
Let 
be the set of rational numbers. A function 
is called 
if the following property holds: for every 
,
![\[f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y).\]](http://latex.artofproblemsolving.com/9/d/b/9db4cc2a779c712b20aac279c97fbd66e821fe2a.png)
Show that there exists an integer 
such that for any aquaesulian function 
there are at most different rational numbers of the form 
for some rational number 
, and find the smallest possible value of .
