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

2004 Germany Team Selection Test, 1
Tags: closed formula , functional equation , function
A function $f$ satisfies the equation \[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\] for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.

2017 Federal Competition For Advanced Students, P2, 1
Tags: algebra , functional equation
Let $\alpha$ be a fixed real number. Find all functions $f:\mathbb R \to \mathbb R$ such that $$f(f(x + y)f(x - y)) = x^2 + \alpha yf(y)$$for all $x,y \in \mathbb R$. [i]Proposed by Walther Janous[/i]

1988 Greece National Olympiad, 1
Tags: function , algebra , functional equation
Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satidfy : $$2f(x+y+xy)= a f(x)+ bf(y)+f(xy)$$ for any $x,y \in\mathbb{R}$ όπου $a,b\in\mathbb{R}$ with $a^2-a\ne b^2-b$

1979 IMO Longlists, 27
Tags: algebra , binary representation , function , functional equation
For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by \[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\] Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.

2001 Estonia Team Selection Test, 3
Tags: algebra , functional equation , functional
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

2004 IMO Shortlist, 3
Tags: divisibility , function , number theory , algebra , functional equation
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

2021 Azerbaijan IMO TST, 3
Tags: nonnegative integers , number theory , functional equation
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]

VMEO II 2005, 7
Tags: algebra , functional equation , function , polynomial
Find all function $f:[0,\infty )\to\mathbb{R}$ such that $f$ is monotonic and \[ [f(x)+f(y)]^2=f(x^2-y^2)+f(2xy) \] for all $x\geq y\geq 0$

2012 Baltic Way, 5
Tags: algebra , functional equation
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which \[f(x + y) = f(x - y) + f(f(1 - xy))\] holds for all real numbers $x$ and $y$.

2024 Moldova Team Selection Test, 9
Tags: algebra , functional equation
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$.

2019 Iran MO (3rd Round), 2
Tags: functional equation
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ : $$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$. [i]Proposed by Amirhossein Zolfaghari [/i]

1980 IMO Longlists, 7
Tags: functional equation , function , algebra
The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

1977 IMO, 3
Tags: functional inequality , functional equation , function , algebra
Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$

2019 VJIMC, 2
Tags: functional equation , real analysis
Find all twice differentiable functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f''(x) \cos(f(x))\geq(f'(x))^2 \sin(f(x)) $$ for every $x\in \mathbb{R}$. [i]Proposed by Orif Ibrogimov (Czech Technical University of Prague), Karim Rakhimov (University of Pisa)[/i]

2022 Kosovo National Mathematical Olympiad, 2
Tags: functional equation , function
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(f(x-y)-yf(x))=xf(y).$$

2007 Grigore Moisil Intercounty, 3
Tags: injectivity , function , algebra , functional equation
Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.

2019 Pan-African Shortlist, A3
Tags: function , algebra , functional equation
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y) $$ for all real numbers $x$ and $y$.

2019 Romanian Master of Mathematics, 5
Tags: functional equation , fe , algebra
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying \[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\] for all real numbers $x$ and $y$.

2013 IMO Shortlist, N6
Tags: number theory , function , functional equation
Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying \[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \] for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)

2010 ELMO Shortlist, 3
Tags: functional equation , max and min
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$. [i]George Xing.[/i]

1957 Miklós Schweitzer, 5
Tags: functional equation
[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]

2015 Thailand Mathematical Olympiad, 9
Tags: functional , algebra , functional equation
Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$

1998 Croatia National Olympiad, Problem 3
Tags: fe , combinatorics , functional equation
Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?

2016 Bosnia and Herzegovina Team Selection Test, 6
Tags: functional equation
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 $x,y\in\mathbb{Z}$.

2017 Thailand TSTST, 3
Tags: functional equation , function , algebra
Let $f$ be a function on a set $X$. Prove that $$f(X-f(X))=f(X)-f(f(X)),$$ where for a set $S$, the notation $f(S)$ means $\{f(a) | a \in S\}$.