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: 196

1999 All-Russian Olympiad Regional Round, 9.7
Tags: number theory , Divisors
Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number $12$ has two prime factors: $2$ and $3$.)

2024 Dutch IMO TST, 1
Tags: number theory , number theory proposed , Divisors , coprime
For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

2009 Flanders Math Olympiad, 2
Tags: number theory , Divisors
A natural number has four natural divisors: $1$, the number itself, and two real divisors. That number plus $9$ is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.

1993 Tournament Of Towns, (381) 3
Tags: number theory , Divisors
A natural number $A$ is given. One may add to it one of its divisors $d$ ($1 < d < A$). One may then repeat this operation with the new number $A + d$ and so on. Prove that starting from $A = 4$ one can get any composite number by these operations. (M Vyalyi)

2003 France Team Selection Test, 3
Tags: algebra , number theory , primes , Divisors , IMO Shortlist
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

1995 Grosman Memorial Mathematical Olympiad, 1
Tags: Divisors , number theory
Positive integers $d_1,d_2,...,d_n$ are divisors of $1995$. Prove that there exist $d_i$ and $d_j$ among them, such the denominator of the reduced fraction $d_i/d_j$ is at least $n$

2024 Baltic Way, 18
Tags: number theory , number theory proposed , Sequence , Divisors , prime numbers , Bound
An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.

2018 Estonia Team Selection Test, 8
Tags: Divisors , number theory
Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors $1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.

2004 IMO Shortlist, 1
Tags: number theory , Divisors , equation , IMO Shortlist
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

2025 Euler Olympiad, Round 2, 1
Tags: Euler Olympiad , number theory , Divisors , relatively prime , prime numbers
Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined. [b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers. [b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers. [b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$. [i]Proposed by Stijn Cambie, Belgium[/i]

2019 Regional Competition For Advanced Students, 4
Tags: number theory , Divisors
Find all natural numbers $n$ that are smaller than $128^{97}$ and have exactly $2019$ divisors.

2018 Danube Mathematical Competition, 1
Tags: number theory , Divisors , Composite
Find all the pairs $(n, m)$ of positive integers which fulfil simultaneously the conditions: i) the number $n$ is composite; ii) if the numbers $d_1, d_2, ..., d_k, k \in N^*$ are all the proper divisors of $n$, then the numbers $d_1 + 1, d_2 + 1, . . . , d_k + 1$ are all the proper divisors of $m$.

2024 Kyiv City MO Round 1, Problem 5
Tags: number theory , Divisors
Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold: $$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$ [i]Proposed by Mykyta Kharin[/i]

2021 Bangladeshi National Mathematical Olympiad, 7
Tags: number theory , function , Divisors , Perfect Squares , perfect cube
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.

2020 AIME Problems, 4
Tags: AMC , AIME , AIME I , 2020 AIME I , 2020 AMC , Divisors
Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020$, and when the last four digits are removed, the result is a divisor of $N$. For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020$. Find the sum of all the digits of all the numbers in $S$. For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total.

2018 Polish Junior MO Finals, 3
Tags: combinatorics , Coloring , number theory , Divisors , Poland
Let $n$ be a positive integer. Each number $1, 2, ..., 1000$ has been colored with one of $n$ colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number $n$ for which it is possible.

2022 Mexico National Olympiad, 3
Tags: music , Divisors
Let $n>1$ be an integer and $d_1<d_2<\dots<d_m$ the list of its positive divisors, including $1$ and $n$. The $m$ instruments of a mathematical orchestra will play a musical piece for $m$ seconds, where the instrument $i$ will play a note of tone $d_i$ during $s_i$ seconds (not necessarily consecutive), where $d_i$ and $s_i$ are positive integers. This piece has "sonority" $S=s_1+s_2+\dots s_n$. A pair of tones $a$ and $b$ are harmonic if $\frac ab$ or $\frac ba$ is an integer. If every instrument plays for at least one second and every pair of notes that sound at the same time are harmonic, show that the maximum sonority achievable is a composite number.

2024 Baltic Way, 19
Tags: number theory proposed , Divisors , construction , Fractions
Does there exist a positive integer $N$ which is divisible by at least $2024$ distinct primes and whose positive divisors $1 = d_1 < d_2 < \ldots < d_k = N$ are such that the number \[ \frac{d_2}{d_1}+\frac{d_3}{d_2}+\ldots+\frac{d_k}{d_{k-1}} \] is an integer?

2007 Indonesia TST, 2
Tags: prime divisors , Divisors , number theory
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

2005 India IMO Training Camp, 2
Tags: number theory , Divisors , equation , IMO Shortlist
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

1998 IMO Shortlist, 6
Tags: number theory , Divisors , Number theoretic functions , prime factorization , IMO , IMO 1998 , IMO Shortlist
For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.