Olympiad problems

Russian TST 2022, P2

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(xy+f(x))+f(y)=xf(y)+f(x+y),\]for all real numbers $x,y$.

1999 IMO Shortlist, 5

Tags: algebra , functional equation

Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that \[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\] for all $x,y \in \mathbb{R} $.

2023 Thailand Online MO, 5

Tags: number theory , arithmetic function , number theoretic functions , functional equation

For each positive integer $k$, let $d(k)$ be the number of positive divisors of $k$ and $\sigma(k)$ be the sum of positive divisors of $k$. Let $\mathbb N$ be the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb N$ such that \begin{align*} f(d(n+1)) &= d(f(n)+1)\quad \text{and} \\ f(\sigma(n+1)) &= \sigma(f(n)+1) \end{align*} for all positive integers $n$.

2018 USA TSTST, 1

Tags: functional equation , algebra

As usual, let ${\mathbb Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : {\mathbb Z}[x] \to {\mathbb Z}$ such that for any polynomials $p,q \in {\mathbb Z}[x]$, [list] [*]$\theta(p+1) = \theta(p)+1$, and [*]if $\theta(p) \neq 0$ then $\theta(p)$ divides $\theta(p \cdot q)$. [/list] [i]Evan Chen and Yang Liu[/i]

2018 Abels Math Contest (Norwegian MO) Final, 4

Tags: algebra , sum , binomial , functional equation

Find all polynomials $P$ such that $P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$ $=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$ for all real numbers $x$.

2010 IMAC Arhimede, 2

Tags: functional equation , algebra

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}$

2011 Brazil Team Selection Test, 4

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]

2017-IMOC, A7

Tags: iteration , algebra , functional equation , function

Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies \[ f^n(n) = n + k \] for all $n \in \mathbb{N}$

2021 ISI Entrance Examination, 2

Tags: functional equation , isi entrance

Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \neq 0 = f(1)$. Assume also that $f$ satisfies equations [b](A)[/b] and [b](B)[/b] below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\ f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)} \end{eqnarray*} for all integers $x,y$. [b](i)[/b] Determine explicitly the set $\big\{f(a)~:~a\in\mathbb{Z}\big\}$. [b](ii)[/b] Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set $\big\{b~:~f(b) \neq 0\big\}$ is infinite.

2019 Brazil Team Selection Test, 1

Tags: functional equation , function , number theory , algebra

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

2024 Iran MO (3rd Round), 1

Tags: functional equation , complex number , fe , function , algebra

Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and: $$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$ Where $\overline{a}$ is the complex conjugate of $a$.

2016 Postal Coaching, 4

Tags: function , functional equation , algebra

Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.

2017 European Mathematical Cup, 1

Tags: functional equation , natural number

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$ holds for all positive integers $x, y$. Proposed by Adrian Beker.

2017 Azerbaijan Team Selection Test, 3

Tags: functional equation , algebra

Find all functions $f : \mathbb R\to\mathbb R $ such that \[f(x+yf(x^2))=f(x)+xf(xy)\] for all real numbers $x$ and $y$.

PEN K Problems, 8

Tags: trigonometry , function , functional equation

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+6f(n)=3f(f(n))+4n+2001.\]

2025 District Olympiad, P3

Tags: function , complex number , algebra , functional equation

Determine all functions $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $$|wf(z)+zf(w)|=2|zw|$$ for all $w,z\in\mathbb{C}$.

2020 Israel Olympic Revenge, P1

Tags: function , algebra , symmetry , functional equation

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$ one has \[f(f(x)+y)=f(x+f(y))\] and in addition the set $f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}$ is a finite set for all $a\in \mathbb{R}$.

KoMaL A Problems 2019/2020, A.756

Tags: functional equation , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$ [i]Based on a problem of Romanian Masters of Mathematics[/i]

1992 IMO Longlists, 35

Tags: algebra , polynomial , functional equation , iteration

Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.

1992 IMO Longlists, 51

Tags: functional equation , algebra , polynomial , number theory

Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose \[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\] Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$

2017 Taiwan TST Round 2, 1

Tags: functional equation , algebra

Determine all surjective functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right) $$ for all $ x,y,z $ in $ \mathbb{Z} $

2014 Serbia National Math Olympiad, 1

Tags: functional equation , function

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$ hold: $$f(xf(y)-yf(x))=f(xy)-xy$$ [i]Proposed by Dusan Djukic[/i]

2021 Nordic, 2

Tags: algebra , functional equation

Find all functions $f:R->R$ satisfying that for every $x$ (real number): $f(x)(1+|f(x)|)\geq x \geq f(x(1+|x|))$

1986 Tournament Of Towns, (116) 4

Tags: functional equation , functional , algebra , function , continuous , analysis

The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous. (A.I. Plotkin, Leningrad)

2011 Germany Team Selection Test, 3

Tags: functional equation , function , algebra

We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$ a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$ b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$