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

2023 IFYM, Sozopol, 4
Tags: functional equation , algebra
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(2x + y + f(x + y)) + f(xy) = y f(x) \] for all real numbers $x$ and $y$.

2023 India IMO Training Camp, 2
Tags: algebra , functional equation , function
Let $g:\mathbb{N}\to \mathbb{N}$ be a bijective function and suppose that $f:\mathbb{N}\to \mathbb{N}$ is a function such that: [list] [*] For all naturals $x$, $$\underbrace{f(\cdots (f}_{x^{2023}\;f\text{'s}}(x)))=x. $$ [*] For all naturals $x,y$ such that $x|y$, we have $f(x)|g(y)$. [/list] Prove that $f(x)=x$. [i]Proposed by Pulkit Sinha[/i]

2011 Ukraine Team Selection Test, 5
Tags: function , algebra , functional equation
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

2022 EGMO, 2
Tags: functional equation , number theory
Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold: (1) $f(ab) = f(a)f(b)$, and (2) at least two of the numbers $f(a)$, $f(b)$, and $f(a+b)$ are equal.

2016 Switzerland - Final Round, 10
Tags: functional , functional equation , algebra
Find all functions $f : R \to R$ such that for all $x, y \in R$: $$f(x + yf(x + y)) = y^2 + f(xf(y + 1)).$$

2016 Romanian Master of Mathematics Shortlist, A1
Tags: functional equation , algebra , functional equation in n
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

2004 Germany Team Selection Test, 1
Tags: function , closed formula , functional equation
A function $f$ satisfies the equation \[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\] for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.

2019 USAJMO, 2
Tags: function , algebra , functional equation
Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying \[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \] for all integers $x$. [i]Proposed by Ankan Bhattacharya[/i]

2017 Puerto Rico Team Selection Test, 1
Tags: functional , functional equation , sum , algebra
Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$.

1976 Bulgaria National Olympiad, Problem 2
Tags: polynomial , functional equation , fe , algebra
Find all polynomials $p(x)$ satisfying the condition: $$p(x^2-2x)=p(x-2)^2.$$

PEN K Problems, 10
Tags: induction , function , functional equation
Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $n\in \mathbb{N}_{0}$: \[f(m+f(n))=f(f(m))+f(n).\]

2014 Taiwan TST Round 2, 2
Tags: algebra , functional equation , function
Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation \[ f(f(f(n))) = f(n+1 ) +1 \] for all $ n\in \mathbb{Z}_{\ge 0}$.

2004 Switzerland - Final Round, 4
Tags: algebra , functional equation , functional
Determine all functions $f : R \to R$ such that for all $x, y \in R$ holds $$f(xf(x) + f(y)) = y + f(x)^2$$

MathLinks Contest 3rd, 2
Tags: algebra , functional equation , functional , number theory
Find all functions $f : \{1, 2, ... , n,...\} \to Z$ with the following properties (i) if $a, b$ are positive integers and $a | b$, then $f(a) \ge f(b)$; (ii) if $a, b$ are positive integers then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.

2012 IMO Shortlist, A6
Tags: function , algebra , combinatorics , functional equation , arrows , induction
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded. [i]Proposed by Palmer Mebane, United States[/i]

2025 Ukraine National Mathematical Olympiad, 11.3
Tags: functional equation , functional inequality
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for any real numbers \(x\) and \(y\), the following inequality holds: \[ f\left(x^2+2y f(x)\right) + (f(y))^2 \leq f\left((x+y)^2\right) \] [i]Proposed by Anton Trygub[/i]

2019 Saudi Arabia Pre-TST + Training Tests, 4.2
Tags: functional , algebra , functional equation
Find all functions $f : R^2 \to R$ that for all real numbers $x, y, z$ satisfies to the equation $f(f(x,z), f(z, y))= f(x, y) + z$

2018 Federal Competition For Advanced Students, P2, 1
Tags: algebra , functional equation
Let $a \ne 0$ be a real number. Find all functions $f : R_{>0}\to R_{>0}$ with $$f(f(x) + y) = ax + \frac{1}{f\left(\frac{1}{y}\right)}$$ for all $x, y \in R_{>0}$. [i](Proposed by Walther Janous)[/i]

2012 Cuba MO, 7
Tags: algebra , functional equation , functional
Find all the functions $f : R\to R$ such that $f(x^2 + f(y)) = y - x^2$ for all $x, y$ reals.

2016 239 Open Mathematical Olympiad, 7
Tags: algebra , functional equation
Find all functions $f:\mathbb{R^+}\to\mathbb{R^+}$ satisfying$$f(xy+x+y)=(f(x)-f(y))f(y-x-1)$$ for all $x>0, y>x+1$.

Russian TST 2017, P2
Tags: putnam analysis , functional equation
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$

2018 Iran MO (2nd Round), 4
Tags: function , functional equation , algebra
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that: $$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$ for all reals $x, y $.

2016 Saudi Arabia GMO TST, 2
Tags: algebra , functional , functional equation
Find all functions $f : Z \to Z$ such that $f (2m + f (m) + f (m)f (n)) = nf (m) + m$ for any integers $m, n$

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 $y$.

2020 Indonesia MO, 7
Tags: functional equation , polynomial , floor function , algebra
Determine all real-coefficient polynomials $P(x)$ such that \[ P(\lfloor x \rfloor) = \lfloor P(x) \rfloor \]for every real numbers $x$.