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

1990 IMO Longlists, 93
Tags: function , algebra , functional equation , number theory
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{}$.

2016 Bundeswettbewerb Mathematik, 3
Tags: function , algebra , functional equation
Find all functions $f$ that is defined on all reals but $\tfrac13$ and $- \tfrac13$ and satisfies \[ f \left(\frac{x+1}{1-3x} \right) + f(x) = x \] for all $x \in \mathbb{R} \setminus \{ \pm \tfrac13 \}$.

2015 IFYM, Sozopol, 6
Tags: algebra , functional equation
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that for $\forall$ $x,y\in \mathbb{R}$ : $f(x+f(x+y))+xy=yf(x)+f(x)+f(y)+x$.

2023 Austrian MO National Competition, 1
Tags: algebra , functional equation
Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

2024 Middle European Mathematical Olympiad, 2
Tags: function , algebra , functional equation
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[yf(x+1)=f(x+y-f(x))+f(x)f(f(y))\] for all $x,y \in \mathbb{R}$.

2017 QEDMO 15th, 8
Tags: algebra , functional , functional equation
For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?

2004 Estonia Team Selection Test, 1
Tags: polynomial , functional equation , functional , algebra
Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.

2013 Romanian Master of Mathematics, 2
Tags: function , floor function , induction , functional equation , algebra
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?

2024 China Team Selection Test, 5
Tags: algebra , functional equation
Find all functions $f:\mathbb N_+\to \mathbb N_+,$ such that for all positive integer $a,b,$ $$\sum_{k=0}^{2b}f(a+k)=(2b+1)f(f(a)+b).$$ [i]Created by Liang Xiao, Yunhao Fu[/i]

2016 Germany Team Selection Test, 2
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 $x,y\in\mathbb{Z}$.

2018 USA Team Selection Test, 2
Tags: function , algebra , functional equation , probability , random walk
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$, \[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\] [i]Proposed by Yang Liu and Michael Kural[/i]

2024 Azerbaijan IZhO TST, 1
Tags: functional equation , algebra
Let $\alpha\neq0$ be a real number. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x;y\in\mathbb{R}$

2019 SG Originals, Q1
Tags: algebra , functional equation
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 $x,y,z\in\mathbb{R}$.

2016 Romania Team Selection Test, 2
Tags: function , algebra , number theory , functional equation
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$

1997 Croatia National Olympiad, Problem 3
Tags: functional equation , fe , number theory , algebra
Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and $$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$ (a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$. (b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$. (c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.

2006 Flanders Math Olympiad, 4
Tags: function , calculus , algebra , functional equation , system of equations
Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that \[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]

1990 IMO Longlists, 22
Tags: algebra , polynomial , arithmetic sequence , number theory , functional equation , divisibility
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.$

2016 Brazil Team Selection Test, 1
Tags: circumcircle , functional equation , right triangle , reflection , geometry
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

1972 IMO Longlists, 7
Tags: function , algebra , functional equation , functional inequality
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

2021 Balkan MO Shortlist, A3
Tags: functional equation , algebra
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$. [i]Proposed by Athanasios Kontogeorgis, Greece[/i]

2005 Taiwan TST Round 3, 1
Tags: function , number theory , algebra , divisibility , 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]

2015 Bulgaria National Olympiad, 4
Tags: functional equation , algebra
Find all functions $f:\mathbb{R^+}\to\mathbb {R^+} $ such that for all $x,y\in R^+$ the followings hold: $i) $ $f (x+y)\ge f (x)+y $ $ii) $ $f (f (x))\le x $

2007 Balkan MO, 2
Tags: algebra , functional equation
Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.

2022 Balkan MO Shortlist, A1
Tags: algebra , functional equation
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[f(x(x + f(y))) = (x + y)f(x),\] for all $x, y \in\mathbb{R}$.

2017 Czech And Slovak Olympiad III A, 3
Tags: functional equation , functional , algebra
Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$