Olympiad problems

2009 Belarus Team Selection Test, 3

a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$? b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$? I. Voronovich

2018 Swedish Mathematical Competition, 2

Tags: functional equation , functional , algebra

Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$. (Here $\sqrt[3]{x}$ is defined all over $R$.)

2014 Swedish Mathematical Competition, 3

Tags: algebra , functional equation , functional

Determine all functions $f: \mathbb R \to \mathbb R$, such that $$ f (f (x + y) - f (x - y)) = xy$$ for all real $x$ and $y$.

2021 Dutch IMO TST, 3

Tags: algebra , functional equation , functional

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

2004 Swedish Mathematical Competition, 3

Tags: functional equation , functional , algebra

A function $f$ satisfies $f(x)+x f(1-x) = x^2$ for all real $x$. Determine $f$ .

2021 Austrian MO National Competition, 4

Tags: functional equation , algebra , functional

Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

1998 Switzerland Team Selection Test, 1

Tags: algebra , periodic , functional , function

A function $f : R -\{0\} \to R$ has the following properties: (i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$, (ii) $f$ takes the value $\frac12$ at least once. Determine $f(-1)$. Prove that $f$ is a periodic function

2018 NZMOC Camp Selection Problems, 10

Tags: functional equation , functional , algebra

Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.

2003 Switzerland Team Selection Test, 10

Tags: functional equation , algebra , functional

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

2008 Thailand Mathematical Olympiad, 6

Tags: functional inequality , algebra , inequalities , functional

Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$. Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.

2015 Estonia Team Selection Test, 5

Tags: functional , functional equation , algebra

Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.

1994 All-Russian Olympiad Regional Round, 11.6

Tags: algebra , functional

Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$ for all $x \ne 1$.

2016 Costa Rica - Final Round, F3

Tags: functional equation , digit , algebra , functional

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

2009 Switzerland - Final Round, 6

Tags: algebra , functional equation , functional

Find all functions $f : R_{>0} \to R_{>0}$, which for all $x > y > z > 0$ is the following equation holds $$f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz.$$

1996 Abels Math Contest (Norwegian MO), 4

Tags: function , functional , algebra , functional equation

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 $f(1995)$.

2016 Belarus Team Selection Test, 2

Tags: functional , functional equation , algebra

Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions: 1) $f(f(x)) =xf(x)-ax$ for all real $x$ 2) $f$ is not constant 3) $f$ takes the value $a$

1995 Israel Mathematical Olympiad, 8

Tags: functional equation , functional , algebra , function

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2015 Costa Rica - Final Round, 3

Tags: algebra , fractional part , functional , functional equation

Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$ ii) $f (f(x)) = f(x)$ where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.

2009 Belarus Team Selection Test, 3

2018 Swedish Mathematical Competition, 2

2014 Swedish Mathematical Competition, 3

2021 Dutch IMO TST, 3

2004 Swedish Mathematical Competition, 3

2021 Austrian MO National Competition, 4

1998 Switzerland Team Selection Test, 1

2018 NZMOC Camp Selection Problems, 10

2003 Switzerland Team Selection Test, 10

2008 Thailand Mathematical Olympiad, 6

2015 Estonia Team Selection Test, 5

1994 All-Russian Olympiad Regional Round, 11.6

2016 Costa Rica - Final Round, F3

2009 Switzerland - Final Round, 6

1996 Abels Math Contest (Norwegian MO), 4

2016 Belarus Team Selection Test, 2

1995 Israel Mathematical Olympiad, 8

2015 Costa Rica - Final Round, 3

2021-IMOC qualification, A3

2022 Saudi Arabia BMO + EGMO TST, 2.4

1994 Italy TST, 3

VMEO III 2006 Shortlist, A5

1986 Poland - Second Round, 1

2011 QEDMO 9th, 7

2021 Federal Competition For Advanced Students, P2, 4