Difference between revisions of "2024 IMO Problems/Problem 6"
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Let <math>\mathbb{Q}</math> be the set of rational numbers. A function <math>f: \mathbb{Q} \to \mathbb{Q}</math> is called <math>\emph{aquaesulian}</math> if the following property holds: for every <math>x,y \in \mathbb{Q}</math>, | Let <math>\mathbb{Q}</math> be the set of rational numbers. A function <math>f: \mathbb{Q} \to \mathbb{Q}</math> is called <math>\emph{aquaesulian}</math> if the following property holds: for every <math>x,y \in \mathbb{Q}</math>, | ||
<cmath> f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). </cmath>Show that there exists an integer <math>c</math> such that for any aquaesulian function <math>f</math> there are at most <math>c</math> different rational numbers of the form <math>f(r) + f(-r)</math> for some rational number <math>r</math>, and find the smallest possible value of <math>c</math>. | <cmath> f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). </cmath>Show that there exists an integer <math>c</math> such that for any aquaesulian function <math>f</math> there are at most <math>c</math> different rational numbers of the form <math>f(r) + f(-r)</math> for some rational number <math>r</math>, and find the smallest possible value of <math>c</math>. | ||
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| + | ==Video Solution== | ||
| + | https://youtu.be/7h3gJfWnDoc | ||
Revision as of 02:33, 20 July 2024
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 .
