Found problems: 1513
The Golden Digits 2024, P2
Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$.
[i]Proposed by Andrei Vila[/i]
2016 Estonia Team Selection Test, 3
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
2018 India National Olympiad, 6
Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that
$\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$;
$\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$.
Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.
2017 Greece Team Selection Test, 3
Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that:
$f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$
and $g(1)=-8$
2011 QEDMO 9th, 7
Find all functions $f: R\to R$, such that $f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)$ for all $x, y \in R$.
2018 Belarusian National Olympiad, 11.1
Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality
$$
f(f(x))=x^2f(x)+ax^2
$$
for all real $x$.
2016 Greece Team Selection Test, 3
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
1992 IMO Longlists, 35
Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.
1999 Mongolian Mathematical Olympiad, Problem 6
Let $f$ be a map of the plane into itself with the property that if $d(A,B)=1$, then $d(f(A),f(B))=1$, where $d(X,Y)$ denotes the distance between points $X$ and $Y$. Prove that for any positive integer $n$, $d(A,B)=n$ implies $d(f(A),f(B))=n$.
2020 Serbia National Math Olympiad, 5
For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions:
$(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$.
$(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$.
Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.
2007 Estonia Team Selection Test, 5
Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.
2016 Korea Winter Program Practice Test, 3
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following.
$f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$
1957 Miklós Schweitzer, 5
[b]5.[/b] Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases:
(a) $x,y,z$ are arbitrary non-zero real numbers;
(b) $a<x,y,z<b (1<a^{3}<b)$.
[b](R. 13)[/b]
2002 IMO Shortlist, 1
Find all functions $f$ from the reals to the reals such that
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
2019 Greece Team Selection Test, 4
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.
2019 Danube Mathematical Competition, 2
Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation
$$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$
for any real numbers $ x,y. $
2010 Saudi Arabia IMO TST, 2
Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$.
Note: $N = \{0,1,2,...\}$
1977 IMO Longlists, 24
Determine all real functions $f(x)$ that are defined and continuous on the interval $(-1, 1)$ and that satisfy the functional equation
\[f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).\]
2024 Iran MO (3rd Round), 1
Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and:
$$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$
Where $\overline{a}$ is the complex conjugate of $a$.
2018 Iran Team Selection Test, 1
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions:
a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$
b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval.
[i]Proposed by Navid Safaei[/i]
2011 Belarus Team Selection Test, 3
Find all functions $f: R \to R ,g: R \to R$ satisfying the following equality $f(f(x+y))=xf(y)+g(x)$ for all real $x$ and $y$.
I. Gorodnin
2005 VJIMC, Problem 2
Let $f:A^3\to A$ where $A$ is a nonempty set and $f$ satisfies:
(a) for all $x,y\in A$, $f(x,y,y)=x=f(y,y,x)$ and
(b) for all $x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3\in A$,
$$f(f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3))=f(f(x_1,y_1,z_1),f(x_2,y_2,z_2),f(x_3,y_3,z_3)).$$
Prove that for an arbitrary fixed $a\in A$, the operation $x+y=f(x,a,y)$ is an Abelian group addition.
2023 Dutch IMO TST, 4
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
2025 Korea - Final Round, P2
Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$
[b](Condition)[/b] For all $x, y \in \mathbb{R}$, $$f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$$
2024 Thailand Mathematical Olympiad, 3
Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$ for all positive reals $x$ and $y$.