Olympiad problems

2014 Brazil Team Selection Test, 4

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}$.

2024 Vietnam Team Selection Test, 1

Tags: function , functional equation , polynomial , algebra

Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$ for all reals $x,y$.

2023 Grand Duchy of Lithuania, 1

Tags: algebra , functional equation

Given a non-zero real number $a$. Find all functions $f : R \to R$, such that $$f(f(x + y)) = f(x + y) + f(x)f(y) + axy$$ for all $x, y \in R$.

2013 Iran MO (3rd Round), 4

Tags: function , functional equation , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and \[f(x+f(y)^2 ) = {f(x+y)}^2.\] (25 points)

2012 IMO, 4

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]

2024 India IMOTC, 13

Tags: functional equation , imotc , algebra

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]

2012 Thailand Mathematical Olympiad, 5

Tags: functional equation , functional , algebra

Determine all functions $f : R \to R$ satisfying $f(f(x) + xf(y))= 3f(x) + 4xy$ for all real numbers $x,y$.

1990 Romania Team Selection Test, 3

Tags: functional equation , functional , algebra , polynomial

Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

2010 IFYM, Sozopol, 7

Tags: algebra , functional equation

Does there exist a function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that: $f(f(x))=-x$, for all $x\in \mathbb{R}$?

2025 Bangladesh Mathematical Olympiad, P8

Tags: algebra , functional equation

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.

2022 USA TSTST, 8

Tags: functional equation

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f \colon \mathbb{N} \to \mathbb{Z}$ such that \[\left\lfloor \frac{f(mn)}{n} \right\rfloor=f(m)\] for all positive integers $m,n$. [i]Merlijn Staps[/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]

1968 German National Olympiad, 3

Tags: algebra , functional equation , functional

Specify all functions $y = f(x)$, each in the largest possible domain (within the range of real numbers) of the equation $$a \cdot f(x^n) + f(-x^n) = bx$$ suffice, where $b$ is any real number, $n$ is any odd natural number and $a$ is a real number with $|a| \ne 1$.

2014 Federal Competition For Advanced Students, P2, 2

Tags: algebra , functional equation

Let $S$ be the set of all real numbers greater than or equal to $1$. Determine all functions$ f: S \to S$, so that for all real numbers $x ,y \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled.

2024 PErA, P5

Tags: functional equation , algebra , function

Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(xf(x)+y^2) = x^2+yf(y) \] for any positive reals $x,y$.

2012 Kyrgyzstan National Olympiad, 4

Tags: functional equation , algebra , function

Find all functions $ f:\mathbb{R}\to\mathbb{R} $ such that $ f(f(x)^2+f(y)) = xf(x)+y $,$ \forall x,y\in R $.

2014-2015 SDML (High School), 3

Tags: functional equation , algebra , function

Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$

2004 IMO Shortlist, 6

Tags: function , algebra , calculus , functional equation

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[ f(x^2+y^2+2f(xy)) = (f(x+y))^2. \] for all $x,y \in \mathbb{R}$.

2010 Contests, 2

Tags: algebra , functional equation

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$

2019 Thailand TSTST, 3

Tags: functional equation , algebra , function

Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$, $\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and $\text{(iii)}$ $f(1)-f(0)$ is even.

2019 South Africa National Olympiad, 5

Tags: algebra , functional equation

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $$ f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3 $$ for all integers $a, b, c$.

2003 Federal Math Competition of S&M, Problem 2

Tags: inequalities , functional equation , functional inequality

Let $ f : [0, 1] \to\ R $ be a function such that :- $1.)$ $f(x) \ge 0$ for all $x$ in $[0,1]$ . $2.)$ $f(1) = 1$ . $3.)$ If $x_1 , x_2$ are in $[0,1]$ such that $x_1 + x_2 \le 1$ , then $f(x_1) + f(x_2) \le f(x_1 + x_2)$ . Show that $f(x) \le 2x $ for all $x$ in $ [0,1] $.

2008 IMO Shortlist, 1

Tags: functional equation , algebra

Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]

2009 Dutch IMO TST, 4

Tags: algebra , functional equation , functional equation in z

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

2018 IMO Shortlist, A1

Tags: functional equation , algebra

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$