Found problems: 1513
2019 Brazil Team Selection Test, 1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2018 China Team Selection Test, 6
Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions:
(1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$
(2) For all $n\ge M$, $f(n)\ge 0$
(3) For all $n>m>M$, $n-m|f(n)-f(m)$
Show that $a$ is a perfect $r$-th power.
Russian TST 2021, P2
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
2020 Kosovo Team Selection Test, 1
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$ satisfy,
$$f\left(x+yf(x+y)\right)=y^2+f(x)f(y)$$
[i]Proposed by Dorlir Ahmeti, Kosovo[/i]
2024 Romania National Olympiad, 3
Find the functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$(f(x)-y)f(x+f(y))=f(x^2)-yf(y),$$ for all real numbers $x$ and $y.$
2004 VJIMC, Problem 2
Find all functions $f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}$ such that
$1$. $f(x,0)=f(0,x)=x$ for all $x\in\mathbb R_{\ge0}$,
$2$. $f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\mathbb R_{\ge0}$ and
$3$. there exists a real $k$ such that $f(x+y,x+z)=kx+f(y,z)$ for all $x,y,z\in\mathbb R_{\ge0}$.
2011 IMO Shortlist, 4
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$.
[i]Proposed by Bojan Bašić, Serbia[/i]
2024 Azerbaijan IZhO TST, 1
Let $\alpha\neq0$ be a real number. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$
for all $x;y\in\mathbb{R}$
2015 Indonesia MO Shortlist, A4
Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that
\[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$
and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.
2017 Taiwan TST Round 2, 4
Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\
(1) $f$ has periodic $2017$\\
(2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\
Proposed by William Chao
2021 European Mathematical Cup, 3
Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that
$$x^2-y^2+2y(f(x)+f(y))$$
is a square of an integer for all positive integers $x$ and $y$.
2022 Taiwan TST Round 3, 4
Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the positive integer set $\mathbb{N}$. Determine all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties:
(i) For all $S$, $T \in \mathcal{X}$ with $S\subseteq T$, there holds $f(T) \le f(S)$.
(ii) For all $S$, $T \in \mathcal{X}$, there hold
\[f(S) + f(T) \le f(S + T),\quad f(S)f(T) = f(S\cdot T), \]
where $S + T = \{s + t\mid s\in S, t\in T\}$ and $S \cdot T = \{s\cdot t\mid s\in S, t\in T\}$.
[i]Proposed by Li4, Untro368, and Ming Hsiao.[/i]
2007 France Team Selection Test, 2
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$:
\[f(x-y+f(y))=f(x)+f(y).\]
2021 SAFEST Olympiad, 6
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
2007 Rioplatense Mathematical Olympiad, Level 3, 4
Find all functions $ f:Z\to Z$ with the following property: if $x+y+z=0$, then $f(x)+f(y)+f(z)=xyz.$
2019 Balkan MO Shortlist, A2
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[ f(xy) = yf(x) + x + f(f(y) - f(x)) \]
for all $x,y \in \mathbb{R}$.
2012 IMAC Arhimede, 3
Find all functions $f:Q^+ \to Q^+$ such that for any $x,y \in Q^+$ :
$$y=\frac{1}{2}\left[f\left(x+\frac{y}{x}\right)- \left(f(x)+\frac{f(y)}{f(x)}\right)\right]$$
1997 Croatia National Olympiad, Problem 3
Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and
$$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$
(a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$.
(b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$.
(c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.
2020 IberoAmerican, 5
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xf(x-y))+yf(x)=x+y+f(x^2),$$ for all real numbers $x$ and $y.$
2022 Vietnam National Olympiad, 2
Find all function $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that:
\[f\left(\frac{f(x)}{x}+y\right)=1+f(y), \quad \forall x,y \in \mathbb R^+.\]
2016 Postal Coaching, 2
Determine all functions $f:\mathbb R\to\mathbb R$ such that for all $x, y \in \mathbb R$
$$f(xf(y) - yf(x)) = f(xy) - xy.$$
2019 Teodor Topan, 3
Let be a positive real number $ r, $ a natural number $ n, $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying $ f(rxy)=(f(x)f(y))^n, $ for any real numbers $ x,y. $
[b]a)[/b] Give three distinct examples of what $ f $ could be if $ n=1. $
[b]b)[/b] For a fixed $ n\ge 2, $ find all possibilities of what $ f $ could be.
[i]Bogdan Blaga[/i]
2017 Balkan MO Shortlist, A6
Find all functions $f : \mathbb R\to\mathbb R $ such that \[f(x+yf(x^2))=f(x)+xf(xy)\] for all real numbers $x$ and $y$.
2018 Taiwan TST Round 1, 5
Find all functions $ f: \mathbb{N} \to \mathbb{Z} $ satisfying $$ n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)} $$ holds for all positive integers $ m,n $
2018 International Olympic Revenge, 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that
\[
f(x)^2-f(y)^2=f(x+y)\cdot f(x-y),
\]
for all $x,y\in \mathbb{Q}$.
[i]Proposed by Portugal.[/i]