Olympiad problems

2022 European Mathematical Cup, 3

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$ for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.

2005 Austrian-Polish Competition, 6

Tags: functional equation , cauchy functional equation , sum of powers , algebra

Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$ \[f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}\]

2025 Euler Olympiad, Round 2, 4

Tags: functional equation , cauchy functional equation , multiplicative function , algebra

Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$, $$ f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y), $$ where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$. [I]Proposed by Stijn Cambie, Belgium[/i]

2004 Nicolae Coculescu, 4

Tags: algebra , function , cauchy functional equation , monotone

Let be a function satisfying [url=http://mathworld.wolfram.com/CauchyFunctionalEquation.html]Cauchy's functional equation,[/url] and having the property that it's monotonic on a real interval. Prove that this function is globally monotonic. [i]Florian Dumitrel[/i]

2023 Mongolian Mathematical Olympiad, 1

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

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 $x,y,z$.