This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 35

2014 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: number of divisors , divisor , number theory
El Chapulín observed that the number $2014$ has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus $4$. A number of eight divisors with this unusual property is called the [i]red[/i] number . How many [i]red[/i] numbers smaller than $2014$ exist?

2021 USEMO, 2
Tags: number of divisors , number theory
Find all integers $n\ge1$ such that $2^n-1$ has exactly $n$ positive integer divisors. [i]Proposed by Ankan Bhattacharya [/i]

2019 Germany Team Selection Test, 1
Tags: arithmetic function , number of divisors , divisor , number theory
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.

2019 Brazil Team Selection Test, 1
Tags: number of divisors , arithmetic function , divisor , number theory
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.

2019 Tournament Of Towns, 1
Tags: number of divisors , divisor , factor , prime factorization , inequalities , prime , number theory
Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity a) not greater than the complexity of $n$. b) less than the complexity of $n$. (Boris Frenkin)

2018 China Team Selection Test, 2
Tags: arithmetic series , number of divisors , number theory
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

2011 IMO Shortlist, 1
Tags: algorithm , prime number , divisibility , number of divisors , number theory
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]

2012 Brazil 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]

2019 239 Open Mathematical Olympiad, 2
Tags: number of divisors , divisor , number theory
Is it true that there are $130$ consecutive natural numbers, such that each of them has exactly $900$ natural divisors?

2023 Iran Team Selection Test, 1
Tags: number of divisors , divis , number theory
Suppose that $d(n)$ is the number of positive divisors of natural number $n$. Prove that there is a natural number $n$ such that $$ \forall i\in \mathbb{N} , i \le 1402: \frac{d(n)}{d(n \pm i)} >1401 $$ [i]Proposed by Navid Safaei and Mohammadamin Sharifi [/i]

2019 Thailand TST, 1
Tags: arithmetic function , number of divisors , divisor , number theory
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.

2018 China Team Selection Test, 2
Tags: arithmetic series , number of divisors , number theory
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

2020 Dutch IMO TST, 1
Tags: number of divisors , number theory
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 $k = d (a) = d (b) = d (2a + 3b)$.

2014 Federal Competition For Advanced Students, P2, 1
Tags: number theory , number of divisors , divisor
For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$. For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?

2019 Belarus Team Selection Test, 3.1
Tags: number theory , arithmetic function , number of divisors , divisor
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.

1987 Mexico National Olympiad, 4
Tags: number theory , product , perfect square , number of divisors
Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.

2021 Saudi Arabia IMO TST, 9
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]

2021 Spain Mathematical Olympiad, 2
Tags: number theory , number of divisors
Given a positive integer $n$, we define $\lambda (n)$ as the number of positive integer solutions of $x^2-y^2=n$. We say that $n$ is [i]olympic[/i] if $\lambda (n) = 2021$. Which is the smallest olympic positive integer? Which is the smallest olympic positive odd integer?

1987 Mexico National Olympiad, 2
Tags: number of divisors , number theory
How many positive divisors does number $20!$ have?

Russian TST 2019, P1
Tags: number theory , arithmetic function , number of divisors , divisor
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.

2018 IMO Shortlist, N1
Tags: number theory , arithmetic function , number of divisors , divisor
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.

2019 Estonia Team Selection Test, 9
Tags: number theory , arithmetic function , number of divisors , divisor
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.

2019 Hong Kong TST, 1
Tags: number theory , arithmetic function , number of divisors , divisor
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.

2019 Philippine TST, 4
Tags: arithmetic function , number of divisors , divisor , number theory
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.

2019 Estonia Team Selection Test, 9
Tags: number theory , arithmetic function , number of divisors , divisor
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.