Olympiad problems

1995 Czech and Slovak Match, 2

Find all pairs of functions $f ,g : Z \rightarrow Z $ that satisfy $f (g(x)+y) = g( f (y)+x) $ for all integers $ x,y$ and such that $g(x) = g(y)$ only if $x = y$.

2007 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: functional equation in z , functional equation , algebra

Find all functions $ f:Z\to Z$ with the following property: if $x+y+z=0$, then $f(x)+f(y)+f(z)=xyz.$

2021 Israel TST, 2

Tags: functional equation in z , functional equation , algebra

Find all unbounded functions $f:\mathbb Z \rightarrow \mathbb Z$ , such that $f(f(x)-y)|x-f(y)$ holds for any integers $x,y$.

2020 Dutch IMO TST, 3

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

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

2016 Balkan MO Shortlist, A8

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

Find all functions $f : Z \to Z$ for which $f(g(n)) - g(f(n))$ is independent on $n$ for any $g : Z \to Z$.

2009 Dutch IMO TST, 4

Tags: functional equation , functional equation in z , algebra

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

2009 Dutch IMO TST, 4

Tags: algebra , functional equation , functional equation in z

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

2021 Israel TST, 2

Tags: functional equation , algebra , functional equation in z

Find all unbounded functions $f:\mathbb Z \rightarrow \mathbb Z$ , such that $f(f(x)-y)|x-f(y)$ holds for any integers $x,y$.

2017 Thailand TSTST, 1

Tags: algebra , functional equation in z , functional equation

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

2023 Iberoamerican, 2

Tags: algebra , functional equation in z

Let $\mathbb{Z}$ be the set of integers. Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that: $$2023f(f(x))+2022x^2=2022f(x)+2023[f(x)]^2+1$$ for each integer $x$.

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 $x, y \in Z$