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

2019 ELMO Shortlist, A2
Tags: functional equation , algebra
Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.) [i]Proposed by Sean Li[/i]

2010 IMO, 1
Tags: algebra , fe , functional equation
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

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

2016 Brazil Team Selection Test, 1
Tags: functional equation , right triangle , reflection , geometry , circumcircle
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]

2018 Belarusian National Olympiad, 11.1
Tags: parameterization , functional equation , algebra
Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$.

2009 IMO Shortlist, 7
Tags: algebra , functional equation , function
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

2025 Kosovo National Mathematical Olympiad`, P2
Tags: algebra , real , functional equation
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that $$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$

PEN K Problems, 23
Tags: functional equation , function
Let ${\mathbb Q}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}$ such that \[f(xf(y)) = \frac{f(x)}{y}\] for all $x, y \in{\mathbb Q}^{+}$.

1990 Putnam, B1
Tags: derivative , integration , calculus , functional equation , algebra , function
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]

2016 Brazil Team Selection Test, 5
Tags: functional equation , algebra , function
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$ for all positive real numbers $x, y, z$. [i]Fajar Yuliawan, Indonesia[/i]

2014 Contests, 2
Tags: symmetry , algebra , function , functional equation
Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

2020 Centroamerican and Caribbean Math Olympiad, 3
Tags: algebra , functional equation , function
Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then $$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$

2014 Iran Team Selection Test, 4
Tags: function , functional equation , algebra
Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that $x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]

2022 USAMTS Problems, 2
Tags: algebra , functional equation , function
Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.

2017 Peru IMO TST, 13
Tags: functional equation , algebra
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$, \[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]

Russian TST 2020, P1
Tags: functional equation , algebra
Determine all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ satisfying $xf(xf(2y))=y+xyf(x)$ for all $x,y>0$.

2000 Moldova National Olympiad, Problem 5
Tags: fe , polynomial , functional equation
Prove that there is no polynomial $P(x)$ with real coefficients that satisfies $$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions?

2017 Turkey Team Selection Test, 7
Tags: functional equation , algebra
Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$ depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.

2020 Indonesia MO, 3
Tags: algebra , divisibility , functional equation
The wording is just ever so slightly different, however the problem is identical. Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.

2021 Canadian Mathematical Olympiad Qualification, 1
Tags: functional equation , algebra , polynomial
Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.

1992 IMO Longlists, 71
Tags: number theory , functional equation , relatively prime , algebra , polynomial
Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$. Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \ (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$

2022-IMOC, A6
Tags: algebra , functional equation
Find all functions $f:\mathbb R^+\to \mathbb R^+$ such that $$f(x+y)f(f(x))=f(1+yf(x))$$ for all $x,y\in \mathbb R^+.$ [i]Proposed by Ming Hsiao[/i]

2023 Austrian MO National Competition, 1
Tags: functional equation , algebra
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 $x, y \in \mathbb{R}$.

2018 Silk Road, 2
Tags: functional equation , algebra
Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$ such that for any real number $x$ the equalities are true: $f\left(x+1\right)=1+f(x)$ and $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$ [url=http://matol.kz/comments/3373/show]source[/url]

2010 Philippine MO, 3
Tags: functional equation , function , algebra
Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.