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

1989 IMO Longlists, 29
Tags: algebra , functional equation , complex number , function
Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

Russian TST 2014, P3
Tags: algebra , functional equation
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $f(0) = 0$ and for any real numbers $x, y$ the following equality holds \[f(x^2+yf(x))+f(y^2+xf(y))=f(x+y)^2.\]

2017 Balkan MO Shortlist, A6
Tags: functional equation , algebra
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 $y$.

2019 Greece Team Selection Test, 4
Tags: functional equation , functional , algebra
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 $x,y>0$.

2024 Israel TST, P3
Tags: functional equation , algebra , continuous function , function , positive real
Find all continuous functions $f\colon \mathbb{R}_{>0}\to \mathbb{R}_{\geq 1}$ for which the following equation holds for all positive reals $x$, $y$: \[f\left(\frac{f(x)}{y}\right)-f\left(\frac{f(y)}{x}\right)=xy\left(f(x+1)-f(y+1)\right)\]

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

2008 Dutch IMO TST, 1
Tags: functional equation , functional equation in n , algebra
Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

Russian TST 2016, P3
Tags: function , algebra , functional equation
Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.

2017-IMOC, A4
Tags: fe , functional equation , functional inequality , algebra
Show that for all non-constant functions $f:\mathbb R\to\mathbb R$, there are two real numbers $x,y$ such that $$f(x+f(y))>xf(y)+x.$$

2015 IMO Shortlist, A2
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 $x,y\in\mathbb{Z}$.

1998 Croatia National Olympiad, Problem 3
Tags: combinatorics , fe , 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) $n=1000$?

1969 IMO Shortlist, 8
Tags: function , functional equation , continuous function , algebra
Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.

2006 Brazil National Olympiad, 3
Tags: function , induction , functional equation , algebra
Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that \[f(xf(y)+f(x)) = 2f(x)+xy\] for every reals $x,y$.

1998 Romania Team Selection Test, 2
Tags: algebra , polynomial , functional equation
Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k \plus{} 1\}$, then $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$.

2012 ELMO Shortlist, 8
Tags: function , modular arithmetic , induction , functional equation , algebra
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$. [i]Sammy Luo and Alex Zhu.[/i]

2015 Postal Coaching, Problem 1
Tags: algebra , functional equation
Let $f:\mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}$ be defined by $f(0)=0$, $$f(2n+1)=2f(n)$$ for $n \ge 0$ and $$f(2n)=2f(n)+1$$ for $n \ge 1$ If $g(n)=f(f(n))$, prove that $g(n-g(n))=0$ for all $n \ge 0$.

2018 Azerbaijan BMO TST, 2
Tags: perfect square , functional equation , functional equation in n , number theory
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

2017 Saudi Arabia IMO TST, 2
Tags: algebra , functional equation
Find all $f:\mathbb{R}\to\mathbb{R}$ satisfying: $$f(xf(y)-y)+f(xy-x)+f(x+y)=2xy,\quad\forall x,y\in\mathbb{R}.$$

2022 Kosovo Team Selection Test, 1
Tags: function , functional equation
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]

1987 Austrian-Polish Competition, 3
Tags: sequence , functional , functional equation , limit
A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

1995 Baltic Way, 10
Tags: function , functional equation , algebra
Find all real-valued functions $f$ defined on the set of all non-zero real numbers such that: (i) $f(1)=1$, (ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x,y,x+y$, (iii) $(x+y)\cdot f(x+y)=xy\cdot f(x)\cdot f(y)$ for all non-zero $x,y,x+y$.

2022 Turkey Team Selection Test, 2
Tags: functional equation , functional equation in r , algebra
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 $x, y \in \mathbb{Q^+}$.

2015 Estonia Team Selection Test, 5
Tags: algebra , functional equation , functional
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.

2024 Dutch IMO TST, 2
Tags: function , functional equation , algebra , functional inequality
Find all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}$ with \[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\] for all $x,y,z \in \mathbb{R}_{\ge 0}$.

2019 Slovenia Team Selection Test, 2
Tags: algebra , functional equation
Determine all non-negative real numbers $a$, for which $f(a)=0$ for all functions $f: \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0} $, who satisfy the equation $f(f(x) + f(y)) = yf(1 + yf(x))$ for all non-negative real numbers $x$ and $y$.