Olympiad problems

2019 Belarus Team Selection Test, 8.2

Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions: 1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$; 2. $f$ takes all integer values. [i](I. Voronovich)[/i]

2023 District Olympiad, P1

Tags: functional equation , continuity , algebra

Determine all continuous functions $f:\mathbb{R}\to\mathbb{R}$ for which $f(1)=e$ and \[f(x+y)=e^{3xy}\cdot f(x)f(y),\]for all real numbers $x{}$ and $y{}$.

PEN K Problems, 20

Tags: functional equation , function , induction

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

2022-IMOC, A5

Tags: algebra , functional equation

Find all functions $f:\mathbb R\to \mathbb R$ such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all $x,y,z\in \mathbb R.$

1991 IMO Shortlist, 23

Tags: function , algebra , polynomial , functional equation

Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that [i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$ [i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$

2021 European Mathematical Cup, 3

Tags: functional equation , number theory

Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$x^2-y^2+2y(f(x)+f(y))$$ is a square of an integer for all positive integers $x$ and $y$.

2006 Hong Kong TST., 2

Tags: function , functional equation , hong kong

The function $f(x,y)$, defined on the set of all non-negative integers, satisfies (i) $f(0,y)=y+1$ (ii) $f(x+1,0)=f(x,1)$ (iii) $f(x+1,y+1)=f(x,f(x+1,y))$ Find f(3,2005), f(4,2005)

PEN K Problems, 23

Tags: function , functional equation

Let ${\mathbb Q}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}$ such that \[f(xf(y)) = \frac{f(x)}{y}\] for all $x, y \in{\mathbb Q}^{+}$.

2023 ELMO Shortlist, A1

Tags: polynomial , algebra , functional equation

Find all polynomials $P(x)$ with real coefficients such that for all nonzero real numbers $x$, \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]

STEMS 2021-22 Math Cat A-B, A3 B1

Tags: algebra , functional equation

Find all functions $f :\mathbb{N} \rightarrow \mathbb{N}$ such that $f(m + f(n)f(m)) = nf(m) + m$ holds for all $m,n \in \mathbb{N}$.

2023 Mongolian Mathematical Olympiad, 1

Tags: functional equation , cauchy functional equation , three variables , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ and $h : \mathbb{R}^2 \to \mathbb{R}$ such that \[f(x+y-z)^2=f(xy)+h(x+y+z, xy+yz+zx)\] for all real numbers $x,y,z$.

VI Soros Olympiad 1999 - 2000 (Russia), 10.1

Tags: algebra , functional , functional equation

Find all real functions of a real numbers, such that for any $x$, $y$ and $z$ holds the equality $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$

2005 Federal Math Competition of S&M, Problem 3

Tags: functional equation , fe , algebra , polynomial

Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and $$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.

2022 Saudi Arabia BMO + EGMO TST, 2.4

Tags: algebra , functional equation , functional

Find all functions $f : R \to R$ such that $$2f(x)f(x + y) -f(x^2) =\frac{x}{2}(f(2x) + 4f(f(y)))$$ for all $x, y \in R$.

1978 IMO, 3

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 $f(240).$

2009 Jozsef Wildt International Math Competition, W. 12

Tags: function , functional equation

Find all functions $f: (0, +\infty)\cap\mathbb{Q}\to (0, +\infty)\cap\mathbb{Q}$ satisfying thefollowing conditions: [list=1] [*] $f(ax) \leq (f(x))^a$, for every $x\in (0, +\infty)\cap\mathbb{Q}$ and $a \in (0, 1)\cap\mathbb{Q}$ [*] $f(x+y) \leq f(x)f(y)$, for every $x,y\in (0, +\infty)\cap\mathbb{Q}$ [/list]

2016 Estonia Team Selection Test, 3

Tags: algebra , functional equation , functional

Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

2016 APMO, 5

Tags: function , functional equation , algebra

Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$ for all positive real numbers $x, y, z$. [i]Fajar Yuliawan, Indonesia[/i]

MathLinks Contest 5th, 2.1

Tags: functional equation

For what positive integers $k$ there exists a function $f : N \to N$ such that for all $n \in N$ we have $\underbrace{\hbox{f(f(... f(n)....))}}_{\hbox{k times}} = f(n) + 2$ ?

1996 Estonia Team Selection Test, 3

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$: $(i)$ $f(x)=-f(-x);$ $(ii)$ $f(x+1)=f(x)+1;$ $(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$

2020 BMT Fall, 5

Tags: algebra , function , functional equation

Let $f:\mathbb{R}^+\to \mathbb{R}^+$ be a function such that for all $x,y \in \mathbb{R}+,\, f(x)f(y)=f(xy)+f\left(\frac{x}{y}\right)$, where $\mathbb{R}^+$ represents the positive real numbers. Given that $f(2)=3$, compute the last two digits of $f\left(2^{2^{2020}}\right)$.

2024 Taiwan TST Round 3, N

Tags: functional equation , combinatorics

For each positive integer $k$, define $r(k)$ as the number of runs of $k$ in base-$2$, where a run is a collection of consecutive $0$s or consecutive $1$s without a larger one containing it. For example, $(11100100)_2$ has $4$ runs, namely $111-00-1-00$. Also, $r(0) = 0$. Given a positive integer $n$, find all functions $f : \mathbb{Z} \rightarrow\mathbb{Z}$ such that \[\sum_{k=0}^{2^n-1} 2^{r(k)}f(k+(-1)^{k} x)=(-1)^{x+n}\text{ for all integer $x$.}\] [i]Proposed by YaWNeeT[/i]