Found problems: 1513
2013 Saudi Arabia Pre-TST, 3.1
Let $f : R \to R$ be a function satisfying $f(f(x)) = 4x + 1$ for all real number $x$. Prove that the equation $f(x) = x$ has a unique solution.
1994 Austrian-Polish Competition, 8
Given real numbers $a, b$, find all functions $f: R \to R$ satisfying
$f(x,y) = af (x,z) + bf(y,z)$ for all $x,y,z \in R$.
2001 Nordic, 2
Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.
2015 Switzerland Team Selection Test, 7
Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $f_1, f_2 \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$
$$f_1 \left(f_2 (y)-x\right)+2x=g(x+y)$$
2019 Belarus Team Selection Test, 8.2
Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions:
1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$;
2. $f$ takes all integer values.
[i](I. Voronovich)[/i]
1993 IMO, 5
Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties:
(i) $f(1) = 2$;
(ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.
2019 Belarusian National Olympiad, 11.7
Find all functions $f:\mathbb R\to\mathbb R$ satisfying the equality
$$
f(f(x)+f(y))=(x+y)f(x+y)
$$
for all real $x$ and $y$.
[i](B. Serankou)[/i]
2022 SG Originals, Q2
Find all functions $f$ mapping non-empty finite sets of integers, to integers, such that
$$f(A+B)=f(A)+f(B)$$
for all non-empty sets of integers $A$ and $B$.
$A+B$ is defined as $\{a+b: a \in A, b \in B\}$.
2016 Switzerland - Final Round, 10
Find all functions $f : R \to R$ such that for all $x, y \in R$:
$$f(x + yf(x + y)) = y^2 + f(xf(y + 1)).$$
KoMaL A Problems 2024/2025, A. 895
Let's call a function $f:\mathbb R\to\mathbb R$[i] weakly periodic[/i] if it is continuous and $f(x+1)=f(f(x))+1$ for all $x\in\mathbb R$.
a) Does there exist a weakly periodic function such that $f(x)>x$ for all $x\in\mathbb R$?
b) Does there exist a weakly periodic function such that $f(x)<x$ for all $x\in\mathbb R$?
[i]Proposed by: András Imolay, Budapest[/i]
BIMO 2022, 3
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$
2019 OMMock - Mexico National Olympiad Mock Exam, 3
Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z}\rightarrow \mathbb{Z}$ such that, for any two integers $m, n$, $$f(m^2)+f(mf(n))=f(m+n)f(m).$$
[i]Proposed by Victor Domínguez and Pablo Valeriano[/i]
1985 IMO Longlists, 33
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
2019 Abels Math Contest (Norwegian MO) Final, 3b
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
1996 Abels Math Contest (Norwegian MO), 4
Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$.
Find all possible values of $f(1995)$.
2024 Abelkonkurransen Finale, 2b
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying
\[xf(f(x)+y)=f(xy)+x^2\]
for all $x,y \in \mathbb{R}$.
1992 Poland - First Round, 4
Determine all functions $f: R \longrightarrow R$ such that
$f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$
1990 IMO Longlists, 6
Let function $f : \mathbb Z_{\geq 0}^0 \to \mathbb N$ satisfy the following conditions:
(i) $ f(0, 0, 0) = 1;$
(ii) $f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);$
(iii) when applying above relation iteratively, if any of $x', y', z$' is negative, then $f(x', y', z') = 0.$
Prove that if $x, y, z$ are the side lengths of a triangle, then $\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}$ is not an integer for any integers $k, m > 1.$
2024 IFYM, Sozopol, 3
Find all functions \( f:\mathbb{Z} \to \mathbb{Z} \) such that
\[
f(x + f(y) - 2y) + f(f(y)) = f(x)
\]
for all integers \( x \) and \( y \).
2015 India IMO Training Camp, 2
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
1982 IMO Shortlist, 1
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.
1985 Traian Lălescu, 1.3
Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ with the property that
$$ f\left( p(x)\right) =p\left( f(x)\right) ,\quad\forall x\in\mathbb{Q} , $$
for all integer polynomials $ p. $
2002 All-Russian Olympiad, 1
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
2010 Contests, 1
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
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]