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Found problems: 2

2025 Kosovo National Mathematical Olympiad`, P4
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously (i) For all $m,n \in \mathbb{N}$ we have: $$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$ (ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.
gcd prime solve in natural functional equation number theory factorization functional equations prime factorization prime numbers

2024 Moldova EGMO TST, 2
Solve over non-negative integers the system $$ \begin{cases} x+y+z^2=xyz, \\ z\leq min(x,y). \end{cases} $$
solve in natural number theory algebra