Olympiad problems

2019 Germany Team Selection Test, 1

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2017 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: divisor , number of divisors , number theory

It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$

2023 Czech-Polish-Slovak Junior Match, 2

Tags: number of divisors , number theory

For a positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$. Determine all positive integers $n$ for which $d(n)$ is the second largest divisor of $n$.

2018 Regional Competition For Advanced Students, 4

Tags: number theory , number of divisors , divisor

Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$. Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$. [i]Proposed by Richard Henner[/i]

2020 Dutch IMO TST, 1

Tags: number theory , number of divisors

For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$. Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.

2018 Moldova Team Selection Test, 9

Tags: number theory , number of divisors

The positive integers $a $ and $b $ satisfy the sistem $\begin {cases} a_{10} +b_{10} = a \\a_{11}+b_{11 }=b \end {cases} $ where $ a_1 <a_2 <\dots $ and $ b_1 <b_2 <\dots $ are the positive divisors of $a $ and $b$ . Find $a$ and $b $ .

2019 Mexico National Olympiad, 1

Tags: function , number theory , perfect power , number of divisors

An integer number $m\geq 1$ is [i]mexica[/i] if it's of the form $n^{d(n)}$, where $n$ is a positive integer and $d(n)$ is the number of positive integers which divide $n$. Find all mexica numbers less than $2019$. Note. The divisors of $n$ include $1$ and $n$; for example, $d(12)=6$, since $1, 2, 3, 4, 6, 12$ are all the positive divisors of $12$. [i]Proposed by Cuauhtémoc Gómez[/i]

2012 Belarus Team Selection Test, 1

Tags: algorithm , number theory , prime number , divisibility , number of divisors

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

2020 IMO Shortlist, N6

Tags: number theory , euler s totient function , number of divisors

For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$ for all $n\ge 1$ [i]Cyprus[/i]

2018 Austria Beginners' Competition, 4

Tags: sum of divisors , number of divisors , number theory

For a positive integer $n$ we denote by $d(n)$ the number of positive divisors of $n$ and by $s(n)$ the sum of these divisors. For example, $d(2018)$ is equal to $4$ since $2018$ has four divisors $(1, 2, 1009, 2018)$ and $s(2018) = 1 + 2 + 1009 + 2018 = 3030$. Determine all positive integers $x$ such that $s(x) \cdot d(x) = 96$. (Richard Henner)