Olympiad problems

2009 Belarus Team Selection Test, 3

Find all real numbers $a$ for which there exists a function $f: R \to R$ asuch that $x + f(y) =a(y + f(x))$ for all real numbers $x,y\in R$. I.Voronovich

2016 Thailand Mathematical Olympiad, 3

Tags: functional equation , algebra , functional

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

2005 Chile National Olympiad, 5

Tags: number theory , functional

Compute $g(2005)$ where $g$ is a function defined on the natural numbers that has the following properties: i) $g(1) = 0$ ii) $g(nm) = g(n) + g(m) + g(n)g(m)$ for any pair of integers $n, m$. iii) $g(n^2 + 1) = (g(n) + 1)^2$ for every integer $n$.

1999 Austrian-Polish Competition, 3

Tags: system of equations , functional equation , functional , algebra

Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$

1991 Austrian-Polish Competition, 9

Tags: function , functional , composition , algebra

For a positive integer $n$ denote $A = \{1,2,..., n\}$. Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$. How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$?

2011 Mathcenter Contest + Longlist, 7 sl9

Tags: functional equation , algebra , functional

Find the function $\displaystyle{f : \mathbb{R}-\left\{ 0\,\right\} \rightarrow \mathbb{R} }$ such that $$f(x)+f(1-\frac{1}{x}) = \frac{1}{x},\,\,\, \forall x \in \mathbb{R}- \{ 0, 1\,\}$$ [i](-InnoXenT-)[/i]

2021 Dutch IMO TST, 3

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

1986 Poland - Second Round, 1

Tags: continuous , functional , functional equation , algebra

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

1990 Greece National Olympiad, 3

Tags: function , algebra , inequalities , functional

Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satisfy $y^2f(x)(f(x)-2x)\le (1-xy)(1+xy) $ for any $x,y \in\mathbb{R}$.

2001 Czech And Slovak Olympiad IIIA, 1

Tags: functional equation , functional , polynomial , algebra

Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$

2010 Contests, 1

Tags: algebra , functional , functional equation

Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

2019 Regional Olympiad of Mexico Center Zone, 2

Tags: inequalities , functional inequality , functional , algebra

Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.

2018 Estonia Team Selection Test, 4

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$

1990 Swedish Mathematical Competition, 5

Tags: functional equation , functional , algebra

Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.

2001 Estonia Team Selection Test, 3

Tags: functional , functional equation , algebra

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

2022 Azerbaijan BMO TST, A2

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

2010 Mathcenter Contest, 4

Tags: geometry , functional , convex

Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral. [i](tatari/nightmare)[/i]

1993 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , functional , functional equation

Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy: (a) for all $n,f(n)$ is a positive integer. (b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$. (c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .

1996 Israel National Olympiad, 2

Tags: polynomial , algebra , functional equation , functional

Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$

1998 Switzerland Team Selection Test, 10

Tags: periodic , function , functional

5. Let $f : R \to R$ be a function that satisfies for all $x \in R$ (i) $| f(x)| \le 1$, and (ii) $f\left(x+\frac{13}{42}\right)+ f(x) = f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)$ Prove that $f$ is a periodic function

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 $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$

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 $x, y \in R$.

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 $x, y \in R$.

2005 Cuba MO, 4

Tags: algebra , functional

Determine all functions $f : R_+ \to R$ such that:$$f(x)f(y) = f(xy) + \frac{1}{x} + \frac{1}{y}$$ for all $x, y$ positive reals.

2005 Switzerland - Final Round, 9

Tags: algebra , functional , functional equation

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