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

I Soros Olympiad 1994-95 (Rus + Ukr), 11.5
Tags: functional equation , functional , algebra
Is there a function $f(x)$ defined for all $x$ and such that for some $a$ and all $x$ holds the equality $$f(x) + f(2x^2 - 1) = 2x + a?$$

1993 Bulgaria National Olympiad, 1
Tags: algebra , functional equation , functional
Find all functions $f$ , defined and having values in the set of integer numbers, for which the following conditions are satisfied: (a) $f(1) = 1$; (b) for every two whole (integer) numbers $m$ and $n$, the following equality is satisfied: $$f(m+n)·(f(m)-f(n)) = f(m-n)·(f(m)+ f(n))$$

2023 Macedonian Mathematical Olympiad, Problem 1
Tags: algebra , functional equation
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ we have: $$xf(x+y)+yf(y-x) = f(x^2+y^2)\,.$$ [i]Authored by Nikola Velov[/i]

1998 Nordic, 1
Tags: function , algebra , functional equation , rational function
Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ holds for all rational numbers $x$ and $y$.

2023 Dutch BxMO TST, 2
Tags: algebra , functional inequality , functional equation
Find all functions $f : \mathbb R \to \mathbb R$ for which \[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\] for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!

STEMS 2021-22 Math Cat A-B, B2
Tags: functional equation , algebra
Let $\mathbb{S}$ be the set of all functions $f:\mathbb{Z}\rightarrow \mathbb{R}$. Now, consider the function $g:\mathbb{S} \rightarrow \mathbb{S} ,g(f(x)) = f(x + 1)-f(x)$. Now, we call a function $f \in \mathbb{S}$ good if $g^n(f(x))=0$ for some natural $n$. Prove that if $s \not = t \in S$ are good functions then $s(m)-t(m)$ is 0 for only finitely many $m \in \mathbb{Z}$.

2015 Dutch BxMO/EGMO TST, 5
Tags: functional equation , algebra
Find all functions $f : R \to R$ satisfying $(x^2 + y^2)f(xy) = f(x)f(y)f(x^2 + y^2)$ for all real numbers $x$ and $y$.

2015 Balkan MO Shortlist, A4
Tags: algebra , function , functional equation
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$ (x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$$ (Albania)

2016 Korea Winter Program Practice Test, 3
Tags: functional equation , algebra
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$

1990 IMO Shortlist, 25
Tags: algebra , number theory , functional equation , function
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

2023 Balkan MO Shortlist, A1
Tags: algebra , functional equation
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[xf(x+f(y))=(y-x)f(f(x)).\] [i]Proposed by Nikola Velov, Macedonia[/i]

2019 Nigerian Senior MO Round 4, 1
Tags: function , inequalities , functional equation , number theory
Let $f: N \to N$ be a function satisfying (a) $1\le f(x)-x \le 2019$ $\forall x \in N$ (b) $f(f(x))\equiv x$ (mod $2019$) $\forall x \in N$ Show that $\exists x \in N$ such that $f^k(x)=x+2019 k, \forall k \in N$

2016 Kosovo Team Selection Test, 4
Tags: functional equation
It is given the function $f:\mathbb{R}\rightarrow \mathbb{R}$ fow which $f(1)=1$ and for all $x\in\mathbb{R}$ satisfied $f(x+5)\geq f(x)+5$ and $f(x+1)\leq f(x)+1$ If $g(x)=f(x)-x+1$ then find $g(2016)$ .

2023 Belarusian National Olympiad, 11.4
Tags: algebra , functional equation
Denote by $R_{>0}$ the set of all positive real numbers. Find all functions $f: R_{>0} \to R_{>0}$ such that for all $x,y \in R_{>0}$ the following equation holds $$f(y)f(x+f(y))=f(1+xy)$$

2019 Iran MO (3rd Round), 3
Tags: functional equation , algebra , function
Let $a,b,c$ be non-zero distinct real numbers so that there exist functions $f,g:\mathbb{R}^{+} \to \mathbb{R}$ so that: $af(xy)+bf(\frac{x}{y})=cf(x)+g(y)$ For all positive real $x$ and large enough $y$. Prove that there exists a function $h:\mathbb{R}^{+} \to \mathbb{R}$ so that: $f(xy)+f(\frac{x}{y})=2f(x)+h(y)$ For all positive real $x$ and large enough $y$.

2005 District Olympiad, 4
Tags: function , functional equation , algebra , induction
Let $n\geq 3$ be an integer. Find the number of functions $f:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that \[ f(f(k)) = f^3(k) - 6f^2(k) + 12f(k) - 6 , \ \textrm{ for all } k \geq 1 . \]

2012 Balkan MO, 4
Tags: functional equation , function , inequalities
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.

2008 SEEMOUS, Problem 3
Tags: linear algebra , inequalities , functional equation , fe , matrix
Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy $$f(XY)\le\min\{f(X),f(Y)\}$$for all $X,Y\in\mathcal M_n(\mathbb R)$.

2016 Israel Team Selection Test, 2
Tags: algebra , functional equation
Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying (for all $x,y \in \mathbb{R}$): $f(x+y)^2 - f(2x^2) = f(y-x)f(y+x) + 2x\cdot f(y)$.

2022 District Olympiad, P1
Tags: function , algebra , arithmetic function , functional equation
Let $f:\mathbb{N}^*\rightarrow \mathbb{N}^*$ be a function such that $\frac{x^3+3x^2f(y)}{x+f(y)}+\frac{y^3+3y^2f(x)}{y+f(x)}=\frac{(x+y)^3}{f(x+y)},~(\forall)x,y\in\mathbb{N}^*.$ $a)$ Prove that $f(1)=1.$ $b)$ Find function $f.$

2017 Middle European Mathematical Olympiad, 1
Tags: algebra , functional equation
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying $$f(x^2 + f(x)f(y)) = xf(x + y)$$ for all real numbers $x$ and $y$.

EGMO 2017, 2
Tags: algebra , functional equation , coloring , function
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties: $(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$ $(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$ [i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]

2019 Canadian Mathematical Olympiad Qualification, 1
Tags: functional equation , function , injective function , injective , algebra
A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$. Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$

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]

Istek Lyceum Math Olympiad 2016, 1
Tags: function , functional equation , algebra
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[f(x+y)=f(x-y)+f(f(1-xy))\] holds for all real numbers $x$ and $y$