Olympiad problems

2021 Science ON grade X, 4

Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$ where both sums are taken over the positive divisors of $n$. [i] (Vlad Robu) [/i]

2018 Pan-African Shortlist, A5

Tags: multiplicative function , function , functional equation , number theory , algebra

Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying: [list] [*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$, [*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and [*] $g(x) \neq x$ for $2 \leq x \leq 2018$. [/list] Find the minimum possible value of $g(2)$.

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]

1998 ITAMO, 6

Tags: algebra , increasing function , multiplicative function , function

We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$, multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$. (a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$. (b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$, then $f(n) = n$ for all $n$. (c) Does (b) remain true if the word ”completely” is omitted?

2024 Mexican University Math Olympiad, 3

Tags: multiplicative function , greatest common divisor , euclidean distance , complex number , number theory

Consider a multiplicative function $ f $ from the positive integers to the unit disk centered at the origin, that is, $ f : \mathbb{Z}^+ \to D^2 \subseteq \mathbb{C} $ such that $ f(mn) = f(m)f(n) $. Prove that for every $ \epsilon > 0 $ and every integer $ k > 0 $, there exist $ k $ distinct positive integers $ a_1, a_2, \dots, a_k $ such that $ \text{gcd}(a_1, a_2, \dots, a_k) = k $ and $ d(f(a_i), f(a_j)) < \epsilon $ for all $ i, j = 1, \dots, k $.

2021 Science ON all problems, 4

Tags: functional equation , number theory , multiplicative function , algebra

Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$ where both sums are taken over the positive divisors of $n$. [i] (Vlad Robu) [/i]