Olympiad problems

2019 Iran MO (3rd Round), 3

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 $y$.

OMMC POTM, 2023 2

Tags: algebra , functional equation , function

Find all functions $f$ from the set of reals to itself so that for all reals $x,y,$ $$f(x)f(f(x)+y) = f(x^2) + f(xy).$$ [i]Proposed by Culver Kwan[/i]

1985 Vietnam Team Selection Test, 3

Tags: function , functional equation , algebra

Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.

2024 Korea Winter Program Practice Test, Q7

Tags: functional equation , algebra

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions: [list][*] $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$ [*] For all real number $x$, $f(g(x))=g(f(x))=0$ [*] For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$ [*] For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$ [/list]

2018 International Olympic Revenge, 4

Tags: algebra , functional equation

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that \[ f(x)^2-f(y)^2=f(x+y)\cdot f(x-y), \] for all $x,y\in \mathbb{Q}$. [i]Proposed by Portugal.[/i]

1990 Swedish Mathematical Competition, 5

Tags: functional equation , functional , algebra

Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.

2001 Estonia Team Selection Test, 3

Tags: functional , functional equation , algebra

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

2024 India IMOTC, 13

Tags: functional equation , algebra , imotc

Find all functions $f:\mathbb R \to \mathbb R$ such that \[ xf(xf(y)+yf(x))= x^2f(y)+yf(x)^2, \] for all real numbers $x,y$. [i]Proposed by B.J. Venkatachala[/i]

2004 Spain Mathematical Olympiad, Problem 3

Tags: functional equation , algebra

Represent for $\mathbb {Z}$ the set of all integers. Find all of the functions ${f:}$ $ \mathbb{Z} \rightarrow \mathbb{Z}$ such that for any ${x,y}$ integers, they satisfy: ${f(x + f(y)) = f(x) - y.}$

2010 Philippine MO, 3

Tags: function , algebra , functional equation

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 $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.

2018 Hong Kong TST, 3

Tags: algebra , functional equation , function

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $$f(f(xy-x))+f(x+y)=yf(x)+f(y)$$ for all real numbers $x$ and $y$.

1992 IMO Longlists, 26

Tags: function , algebra , functional equation

Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

2022 Azerbaijan BMO TST, A2

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 $x, y \in R$.

2024 ELMO Shortlist, A7

Tags: functional equation , algebra

For some positive integer $n,$ Elmo writes down the equation \[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\] Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation \[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\] Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$? [i]Srinivas Arun[/i]

2012 IMO Shortlist, A1

Tags: algebra , functional equation

Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: \[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\] (Here $\mathbb{Z}$ denotes the set of integers.) [i]Proposed by Liam Baker, South Africa[/i]

2003 IMO Shortlist, 2

Tags: function , algebra , functional equation

Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that (i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$. [i]Proposed by A. Di Pisquale & D. Matthews, Australia[/i]

2021 Bangladesh Mathematical Olympiad, Problem 12

Tags: algebra , functional equation

A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$.

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]

2020 Caucasus Mathematical Olympiad, 4

Tags: functional equation , number theory , algebra

Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.

1993 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , functional , functional equation

Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy: (a) for all $n,f(n)$ is a positive integer. (b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$. (c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .

2023 India IMO Training Camp, 2

Tags: algebra , functional equation

Let $\mathbb R^+$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ satisfying \[f(x+y^2f(x^2))=f(xy)^2+f(x)\] for all $x,y \in \mathbb{R}^+$. [i]Proposed by Shantanu Nene[/i]

2025 Euler Olympiad, Round 2, 4

Tags: functional equation , cauchy functional equation , multiplicative function , algebra

Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$, $$ f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y), $$ where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$. [I]Proposed by Stijn Cambie, Belgium[/i]

2016 Switzerland Team Selection Test, Problem 9

Tags: function , algebra , functional equation

Find all functions $f : \mathbb{R} \mapsto \mathbb{R} $ such that $$ \left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)$$ for all $x,y \in \mathbb{R}$

1996 Israel National Olympiad, 2

Tags: polynomial , algebra , functional equation , functional

Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$

1992 IMO Shortlist, 2

Tags: quadratic , algebra , functional equation , recurrence relation

Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation: \[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\] Prove that there exists a unique solution of this equation.