Find all functions $f:R_{+}\rightarrow R_{+}$ such that for all $x,y\in R_{+}, f(x)=f(f(f(x))+y)+f(xf(y))f(x+y).$

Asked Oct 07 '23 at 17:20

Active Oct 27 '23 at 21:25

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Let $\mathbb{R}_{+}$ be the set of all positive real numbers. Find all functions $f{:}~ \mathbb{R}_{+} \mapsto \mathbb{R}_{+}$ such that for all $x,y\in R_{+}$, we have $$ f(x)=f(f(f(x))+y)+f(xf(y))f(x+y). $$

This is the third question from USAMO 2022. I tried to solve it but failed because of the annoying $f(f(x))$. However, I couldn’t find an official answer since it is written “Work In Process”.

edited Oct 27 '23 at 21:25

Kevin Dietrich

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asked Oct 07 '23 at 17:20

Arachnephob1a

By assuming $f(x)$ is a polynomial and its degree is $n$, I obtain that $n=n^3$ and thus $n=-1 or 1$. Therefore it has to be $f(x)=ax+b or f(x)=a/x$. But I have no idea how to prove that $f(x)$ is strictly a polynomial. – Arachnephob1a Oct 11 '23 at 03:31
@Arachnephob1a How can $ax+b$ work? The left hand side would be linear in $x$ and the right hand side would be quadratic in $x$. – M W Oct 11 '23 at 03:40
Oh I was just being stupid :) – Arachnephob1a Oct 12 '23 at 01:37
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Search (google) for "USAMO 2022 Solution Notes" by Evan Chen (direct link: https://web.evanchen.cc/exams/USAMO-2022-notes.pdf ). You will find pdf with solution to that problem. I must admit that I don't think I would be able to solve it on my own... – freakish Oct 28 '23 at 15:36

Find all functions $f:R_{+}\rightarrow R_{+}$ such that for all $x,y\in R_{+}, f(x)=f(f(f(x))+y)+f(xf(y))f(x+y).$

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