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

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Found problems: 1513

2002 All-Russian Olympiad, 1
Tags: functional equation , algebra , equation , polynomial , quadratic , ring theory
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.

2009 Jozsef Wildt International Math Competition, W. 12
Tags: function , functional equation
Find all functions $f: (0, +\infty)\cap\mathbb{Q}\to (0, +\infty)\cap\mathbb{Q}$ satisfying thefollowing conditions: [list=1] [*] $f(ax) \leq (f(x))^a$, for every $x\in (0, +\infty)\cap\mathbb{Q}$ and $a \in (0, 1)\cap\mathbb{Q}$ [*] $f(x+y) \leq f(x)f(y)$, for every $x,y\in (0, +\infty)\cap\mathbb{Q}$ [/list]

2005 iTest, 4
Tags: function , algebra , functional equation , functional
The function f is defined on the set of integers and satisfies $\bullet$ $f(n) = n - 2$, if $n \ge 2005$ $\bullet$ $f(n) = f(f(n+7))$, if $n < 2005$. Find $f(3)$.

2003 Cuba MO, 8
Tags: functional equation , complex number , algebra
Find all the functions $f : C \to R^+$ such that they fulfill simultaneously the following conditions: $$(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C$$ $$(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C$$ $$(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C$$

2020 ELMO Problems, P1
Tags: algebra , functional equation
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $$f^{f^{f(x)}(y)}(z)=x+y+z+1$$ for all $x,y,z \in \mathbb{N}$. [i]Proposed by William Wang.[/i]

2008 Iran Team Selection Test, 11
Tags: relatively prime , function , modular arithmetic , number theory , functional equation
$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]

PEN K Problems, 24
Tags: function , induction , functional equation
A function $f$ is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.

1990 IMO Longlists, 22
Tags: functional equation , number theory , arithmetic sequence , algebra , divisibility , polynomial
Let $ f(0) \equal{} f(1) \equal{} 0$ and \[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\] Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$

2002 All-Russian Olympiad, 1
Tags: functional equation , algebra , polynomial , quadratic , ring theory , equation
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.

2019 BAMO, 4
Tags: functional equation , algebra , function
Let $S$ be a finite set of nonzero real numbers, and let $f : S\to S$ be a function with the following property: for each $x \in S$, either $f ( f (x)) = x+ f (x)$ or $f ( f (x)) = \frac{x+ f (x)}{2}$. Prove that $f (x) = x$ for all $x \in S$.

2010 Romania National Olympiad, 4
Tags: induction , function , functional equation , algebra , number theory
Consider the set $\mathcal{F}$ of functions $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ is the set of non-negative integers) having the property that \[f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b.\] a) Determine the set $\{f(1)\mid f\in\mathcal{F}\}$. b) Prove that $\mathcal{F}$ has exactly two elements. [i]Nelu Chichirim[/i]

2005 Iran MO (2nd round), 3
Tags: algebra , functional equation , function
Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all positive real numbers $x$ and $y$, the following equation holds: \[(x+y)f(f(x)y)=x^2f(f(x)+f(y)).\]

2014 Belarus Team Selection Test, 3
Tags: functional equation , algebra
Do there exist functions $f$ and $g$, $f : R \to R$, $g : R \to R$ such that $f(x + f(y)) = y^2 + g(x)$ for all real $x$ and $y$ ? (I. Gorodnin)

1992 IMO Longlists, 48
Tags: function , functional equation , algebra
Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity \[f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+\] Where $\alpha,\beta$ are given real numbers.

2017 Benelux, 1
Tags: functional equation , algebra
Find all functions $f : \Bbb{Q}_{>0}\to \Bbb{Z}_{>0}$ such that $$f(xy)\cdot \gcd\left( f(x)f(y), f(\frac{1}{x})f(\frac{1}{y})\right) = xyf(\frac{1}{x})f(\frac{1}{y}),$$ for all $x, y \in \Bbb{Q}_{>0,}$ where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b.$

BIMO 2022, 3
Tags: functional equation , algebra
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$

2021 Ukraine National Mathematical Olympiad, 4
Tags: functional , algebra , functional equation
Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$ (Vadym Koval)

2024 Austrian MO National Competition, 1
Tags: algebra , parameter , parameterization , functional equation
Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[f(\alpha f(x)+f(y))=\beta x+f(y)\] holds for all real $x$ and $y$. [i](Walther Janous)[/i]

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]

2018 Israel Olympic Revenge, 4
Tags: function , algebra , functional equation
Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that $$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$ for all $f,g:\mathbb R\to\mathbb R$. Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that $$F(f)=\sigma\circ f\circ\sigma$$ for all $f:\mathbb R\to\mathbb R$.

Russian TST 2017, P1
Tags: polynomial , functional equation , sum of digits , number theory
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2008 Ukraine Team Selection Test, 5
Tags: algebra , functional equation
Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/i]

2018 India IMO Training Camp, 3
Tags: algebra , functional equation
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.

2017 QEDMO 15th, 4
Tags: functional equation , functional , algebra
Find all functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.

2020 Estonia Team Selection Test, 3
Tags: functional equation , functional , algebra
Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$