Olympiad problems

2017 Peru IMO TST, 14

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]

2019 Dutch IMO TST, 4

Tags: number theory , function , find all functions , prime number , functional equation

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

1996 Abels Math Contest (Norwegian MO), 4

Tags: functional equation , function , functional , algebra

Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$. Find all possible values of $f(1995)$.

Russian TST 2018, P1

Tags: function , algebra , functional equation , fe

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

2020 Dutch IMO TST, 3

Tags: functional equation , functional equation in z , functional , algebra

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

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 $x, y \in \mathbb{R}$.

2021 Ukraine National Mathematical Olympiad, 4

Tags: functional equation , functional , algebra

Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$ (Vadym Koval)

2001 China Team Selection Test, 3

Tags: function , polynomial , functional equation , algebra

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

PEN K Problems, 26

Tags: functional equation , function , calculus , integration , induction

The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$.

2023 Thailand Online MO, 5

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

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 $n$.

2022 Taiwan TST Round 2, A

Tags: algebra , functional equation , ming

Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying \[f\bigl(x + y^2 f(y)\bigr) = f\bigl(1 + yf(x)\bigr)f(x)\] for any positive reals $x$, $y$, where $\mathbb{R}^+$ is the collection of all positive real numbers. [i]Proposed by Ming Hsiao.[/i]

1996 IMO Shortlist, 8

Tags: function , number theory , algebra , functional equation

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

2012 India National Olympiad, 6

Tags: function , group theory , algebra , functional equation

Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and $(i) f(xy) + f(x)f(y) = f(x) + f(y)$ $(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $ for all $x,y \in \mathbb{Z}$, simultaneously. $(a)$ Find the set of all possible values of the function $f$. $(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find the set of all integers $n$ such that $f(n) \ne 0$.

2021 Romania National Olympiad, 4

Tags: functional equation , algebra

Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that \[f(|x-y|)=|f(x)-f(y)|\] for each $x,y\in\mathbb Z_{\ge 0}$. [i]Andrei Bâra[/i]

1978 IMO Longlists, 24

Tags: algebra , functional equation , sequence , partition

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

2024 Canadian Mathematical Olympiad Qualification, 1

Tags: algebra , functional equation

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

1976 Bulgaria National Olympiad, Problem 2

Tags: polynomial , fe , functional equation , algebra

Find all polynomials $p(x)$ satisfying the condition: $$p(x^2-2x)=p(x-2)^2.$$

2014 Brazil Team Selection Test, 1

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 Costa Rica - Final Round, 4

Tags: algebra , function , functional equation

Maria was a brilliant mathematician who found the following property about her year of birth: if $f$ is a function defined in the set of natural numbers $N = \{0, 1, 2, 3, 4, 5,...\}$ such that $f(1) = 1335$ and $f(n+1) = f(n)-2n+43$ for all $n \in N$, then his year of birth is the maximum value that $f(n)$ can reach when $n$ takes values in $N$. Determine the year of birth of Mary.

2020 Serbia National Math Olympiad, 5

Tags: functional equation , function , algebra , number theory

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.

2024 Baltic Way, 1

Tags: algebra , functional equation

Let $\alpha$ be a non-zero real number. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that \[ xf(x+y)=(x+\alpha y)f(x)+xf(y) \] for all $x,y\in\mathbb{R}$.

2003 Switzerland Team Selection Test, 10

Tags: functional equation , functional , algebra

Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.

1987 IMO Shortlist, 1

Tags: function , algebra , functional equation

Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$. Find $f(1987)$. [i]Proposed by Australia.[/i]

2017 Costa Rica - Final Round, 6

Tags: algebra , functional equation , functional

Let $f:] 0. \infty [ \to R$ be a strictly increasing function, such that $$f (x) f\left(f (x) +\frac{1}{x} \right)= 1.$$ Find $f (1)$.

1998 Poland - Second Round, 1

Tags: function , composite function , functional equation , algebra

Let $A_n = \{1,2,...,n\}$. Prove or disprove: For all integers $n \ge 2$ there exist functions $f,g : A_n \to A_n$ which satisfy $f(f(k)) = g(g(k)) = k$ for $1 \le k \le n$, and $g(f(k)) = k +1$ for $1 \le k \le n -1$.