This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 1513

2020 DMO Stage 1, 3.
Tags: algebra , functional equation
[b]Q.[/b] Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \geqslant x+1, \forall\ x \in \mathbb{R}\quad \text{and}\quad f(x+y) \geqslant f(x) f(y), \forall\ x, y \in \mathbb{R}$$ [i]Proposed by TuZo[/i]

2012 EGMO, 3
Tags: functional equation , algebra , function
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f\left( {yf(x + y) + f(x)} \right) = 4x + 2yf(x + y)\] for all $x,y\in\mathbb{R}$. [i]Netherlands (Birgit van Dalen)[/i]

1995 China National Olympiad, 2
Tags: functional equation , algebra , function
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (1) $f(1)=1$; (2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$; (3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$. Find all solutions of equation $f(k) +f(l)=293$, where $k<l$. ($\mathbb{N}$ denotes the set of all natural numbers).

1988 Austrian-Polish Competition, 4
Tags: algebra , functional equation , functional , increasing
Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$.

2017 Romania EGMO TST, P3
Tags: functional equation , function , algebra
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$

2018 Czech-Polish-Slovak Match, 1
Tags: algebra , functional equation , function
Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i]

1995 IMO Shortlist, 6
Tags: functional equation , function , algebra
Let $ \mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $ f: \mathbb{N} \mapsto \mathbb{N}$ satisfying \[ f(m \plus{} f(n)) \equal{} n \plus{} f(m \plus{} 95) \] for all $ m$ and $ n$ in $ \mathbb{N}.$ What is the value of $ \sum^{19}_{k \equal{} 1} f(k)?$

1995 Grosman Memorial Mathematical Olympiad, 6
Tags: functional , algebra , functional equation
(a) Prove that there is a unique function $f : Q \to Q$ satisfying: (i) $f(q)= 1 + f\left(\frac{q}{1-2q}\right)$ for $0<q< \frac12$ (ii) $f(q)= 1 + f(q-1)$ for $1<q\le 2$ (iii) $f(q)f\left(\frac{1}{q}\right)=1$ for all $q\in Q^+$ (b) For this function $f$ , find all $r\in Q^+$ such that $f(r) = r$

2019 IFYM, Sozopol, 2
Tags: functional equation , number theory
Does there exist a strictly increasing function $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for $\forall$ $n\in \mathbb{N}$: $f(f(f(n)))=n+2f(n)$?

2024 Ukraine National Mathematical Olympiad, Problem 4
Tags: function , functional equation , algebra
Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following: $$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$ [i]Proposed by Mykhailo Shtandenko[/i]

2007 District Olympiad, 3
Tags: algebra , functional equation , function , number theory
Find all functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ f(x)^2+y\vdots x^2+f(y) ,\quad\forall x,y\in\mathbb{N} . $$

2020 Iran Team Selection Test, 4
Tags: algebra , functional equation
Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$. [i]Proposed by Ali Zamani [/i]

2013 Switzerland - Final Round, 4
Tags: functional , functional equation , algebra
Find all functions $f : R_{>0} \to R_{>0}$ with the following property: $$f \left( \frac{x}{y + 1}\right) = 1 - xf(x + y)$$ for all $x > y > 0$ .

2013 ELMO Shortlist, 3
Tags: polynomial , functional equation , algebra
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$. [i]Proposed by Calvin Deng[/i]

2023 District Olympiad, P1
Tags: algebra , functional equation , continuity
Determine all continuous functions $f:\mathbb{R}\to\mathbb{R}$ for which $f(1)=e$ and \[f(x+y)=e^{3xy}\cdot f(x)f(y),\]for all real numbers $x{}$ and $y{}$.

JOM 2025, 3
Tags: functional equation , algebra
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(x)^2+f(2y+1)=x^2+f(y)+y+1\] for all reals $x$, $y$. [i](Proposed by Lim Yun Zhe)[/i]

2023 Azerbaijan BMO TST, 3
Tags: algebra , functional equation
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

2014 Germany Team Selection Test, 2
Tags: functional equation , algebra , number theory
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

2024 USAJMO, 5
Tags: functional equation , function , pointwise trap
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[ f(x^2-y)+2yf(x)=f(f(x))+f(y) \] for all $x,y\in\mathbb{R}$. [i]Proposed by Carl Schildkraut[/i]

1989 Chile National Olympiad, 6
Tags: functional , functional equation , function , algebra
The function $f$, with domain on the set of non-negative integers, is defined by the following : $\bullet$ $f (0) = 2$ $\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value. Determine $f (n)$.

2022 Kosovo Team Selection Test, 1
Tags: functional equation , function
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x^2)+2f(xy)=xf(x+y)+yf(x).$$ [i]Proposed by Dorlir Ahmeti, Kosovo[/i]

2017 Estonia Team Selection Test, 6
Tags: algebra , functional equation
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2023 India IMO Training Camp, 2
Tags: algebra , functional equation
Let $\mathbb R^+$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ satisfying \[f(x+y^2f(x^2))=f(xy)^2+f(x)\] for all $x,y \in \mathbb{R}^+$. [i]Proposed by Shantanu Nene[/i]

2019 SG Originals, Q1
Tags: functional equation , algebra
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(f(x)+x+f(y)f(z))=f(x+zf(x)+zf(y))-xf(z-1)\]for all $x,y,z\in\mathbb{R}$.

2011 Dutch IMO TST, 2
Tags: functional equation , algebra
Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.