Olympiad problems

2015 IFYM, Sozopol, 1

Determine all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy the following equations: a) $f(f(n))=4n+3$ $\forall$ $n \in \mathbb{Z}$; b) $f(f(n)-n)=2n+3$ $\forall$ $n \in \mathbb{Z}$.

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 $x,y$.

2021 Dutch IMO TST, 3

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

2022 JHMT HS, 9

Tags: algebra , functional equation , calculus , integration

There is a unique continuous function $f$ over the positive real numbers satisfying $f(4) = 1$ and \[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \] for all positive $x$. Compute the value of $\int_{0}^{140} (f(x))^3\,dx$.

2020 USEMO, 4

Tags: algebra , functional equation

A function $f$ from the set of positive real numbers to itself satisfies $$f(x + f(y) + xy) = xf(y) + f(x + y)$$ for all positive real numbers $x$ and $y$. Prove that $f(x) = x$ for all positive real numbers $x$.

2023 IFYM, Sozopol, 4

Tags: algebra , functional equation

Find all real numbers $a$ for which there exist functions $f,g: \mathbb{R} \to \mathbb{R}$, where $g$ is strictly increasing, such that $f(1) = 1$, $f(2) = a$, and \[ f(x) - f(y) \leq (x-y)(g(x) - g(y)) \] for all real numbers $x$ and $y$.

2005 India IMO Training Camp, 2

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]

2020 Durer Math Competition Finals, 5

Tags: functional equation , functional , algebra

Let $H = \{-2019,-2018, ...,-1, 0, 1, 2, ..., 2020\}$. Describe all functions $f : H \to H$ for which a) $x = f(x) - f(f(x))$ holds for every $x \in H$. b) $x = f(x) + f(f(x)) - f(f(f(x)))$ holds for every $x \in H$. c) $x = f(x) + 2f(f(x)) - 3f(f(f(x)))$ holds for every $x \in H$. PS. (a) + (b) for category E 1.5, (b) + (c) for category E+ 1.2

2002 Canada National Olympiad, 5

Tags: function , modular arithmetic , algebra , functional equation

Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that \[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \] for all $x$ and $y$ in $\mathbb N$.

2000 Singapore Team Selection Test, 1

Tags: algebra , functional equation , functional

Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

2012 Dutch IMO TST, 5

Tags: functional equation , algebra

Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.

2013 Stars Of Mathematics, 4

Tags: induction , algebra , functional equation , combinatorics

Given a (fixed) positive integer $N$, solve the functional equation \[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\] [i](Dan Schwarz)[/i]

2017 Korea National Olympiad, problem 7

Tags: algebra , functional equation , function , inequalities

Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following. For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$. Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.

2012 Belarus Team Selection Test, 3

Tags: algebra , functional , functional equation

Find all functions $f : Q \to Q$, such that $$f(x + f (y + f(z))) = y + f(x + z)$$ for all $x ,y ,z \in Q$ . (I. Voronovich)

PEN K Problems, 14

Tags: function , inequalities , functional equation

Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that for all $m\in\mathbb{Z}$: [list][*] $f(m+8) \le f(m)+8$, [*] $f(m+11) \ge f(m)+11$.[/list]

2013 USA Team Selection Test, 4

Tags: function , induction , functional equation , combinatorics , arrows , algebra

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]

2023 ELMO Shortlist, A2

Tags: algebra , functional equation

Let $\mathbb R_{>0}$ denote the set of positive real numbers. Find all functions $f:\mathbb R_{>0}\to\mathbb R_{>0}$ such that for all positive real numbers $x$ and $y$, \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\] [i]Proposed by Luke Robitaille[/i]

2011 IMO Shortlist, 3

Tags: function , algebra , functional equation

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

2024 Canadian Mathematical Olympiad Qualification, 1

Tags: algebra , functional equation

Find all functions $f : R \to R$ that satisfy the functional equation $$f(x + f(xy)) = f(x)(1 + y).$$

2023 Dutch BxMO TST, 2

Tags: functional inequality , functional equation , algebra

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!

2022 Singapore MO Open, Q3

Tags: functional equation , number theory , factorial , function

Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

2019 All-Russian Olympiad, 1

Tags: algebra , functional equation

Each point $A$ in the plane is assigned a real number $f(A).$ It is known that $f(M)=f(A)+f(B)+f(C),$ whenever $M$ is the centroid of $\triangle ABC.$ Prove that $f(A)=0$ for all points $A.$

2014 France Team Selection Test, 4

Tags: number theory , algebra , functional equation

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

2022 ELMO Revenge, 4

Tags: functional equation , function , algebra

Find all ordered pairs of integers $(a,b)$ such that there exists a function $f\colon \mathbb{N} \to \mathbb{N}$ satisfying $$f^{f(n)}(n)=an+b$$ For all $n\in \mathbb{N}$.

2017 Ukraine Team Selection Test, 7

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]