Olympiad problems

Kvant 2019, M2562

Each point $A$ in the plane is assigned a real number $f(A).$ It is known that $f(M)=f(A)+f(B)+f(C),$ whenever $M$ is the centroid of $\triangle ABC.$ Prove that $f(A)=0$ for all points $A.$

2014 Contests, 1

Tags: functional equation , functional equation in n , algebra

The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$

2023 Azerbaijan BMO TST, 3

Tags: functional equation , algebra

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

2006 Brazil National Olympiad, 3

Tags: function , induction , functional equation , algebra

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

PEN K Problems, 34

Tags: function , search , functional equation

Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn+m+n)=4f(m)f(n)+f(m)+f(n). \]

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]

1986 Poland - Second Round, 1

Tags: algebra , continuous , functional , functional equation

Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$

2016 Brazil Undergrad MO, 2

Tags: functional equation

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x^2+y^2f(x)) = xf(y)^2-f(x)^2 \] for every $x, y \in \mathbb{R}$

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 $f(240).$

1977 IMO Longlists, 23

Tags: algebra , polynomial , functional equation

For which positive integers $n$ do there exist two polynomials $f$ and $g$ with integer coefficients of $n$ variables $x_1, x_2, \ldots , x_n$ such that the following equality is satisfied: \[\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ? \]

2001 Saint Petersburg Mathematical Olympiad, 11.6

Tags: algebra , functional equation , function

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true: $$f(x+y+f(y))=f(x)+2y$$ [I]proposed by F. Petrov[/i]

2005 Switzerland - Final Round, 9

Tags: algebra , functional , functional equation

Find all functions $f : R^+ \to R^+$ such that for all $x, y > 0$ $$f(yf(x))(x + y) = x^2(f(x) + f(y)).$$

Dumbest FE I ever created, 1.

Tags: function , number theory , functional equation , divisibility , euler s phi function

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$, $$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$

2023 Israel TST, P3

Tags: functional equation , algebra , function

Find all functions $f:\mathbb{Z}\to \mathbb{Z}_{>0}$ for which \[f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1\] holds for any $x,y\in \mathbb{Z}$.

2004 Switzerland - Final Round, 4

Tags: algebra , functional , functional equation

Determine all functions $f : R \to R$ such that for all $x, y \in R$ holds $$f(xf(x) + f(y)) = y + f(x)^2$$

2000 Moldova National Olympiad, Problem 4

Tags: functional equation , fe , polynomial , algebra

Find all polynomials $P(x)$ with real coefficients that satisfy the relation $$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$

1963 Swedish Mathematical Competition., 4

Tags: functional equation , functional , algebra , analysis

Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.

2022 Nordic, 1

Tags: functional equation , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$, for all real $x$.

2010 Bundeswettbewerb Mathematik, 4

Tags: polynomial , function , functional equation , functional , algebra

In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$. (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.