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: 310

2018 India PRMO, 1
Tags: prime , divisor , number theory
A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?

2012 Kyiv Mathematical Festival, 4
Tags: divisor , number theory
Find all positive integers $a, b,c$ greater than $1$, such that $ab + 1$ is divisible by $c, bc + 1$ is divisible by $a$ and $ca + 1$ is divisible by $b$.

2017 Germany, Landesrunde - Grade 11/12, 4
Tags: combinatorics , divisor , number theory
Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.

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 AMC 10, 9
Tags: divisibility , divisor
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers? $\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$

2018 Flanders Math Olympiad, 3
Tags: divisor , sum , number theory
Write down $f(n)$ for the greatest odd divisor of $n \in N_0$. (a) Determine $f (n + 1) + f (n + 2) + ... + f(2n)$. (b) Determine $f(1) + f(2) + f(3) + ... + f(2n)$.

2024 Bulgaria MO Regional Round, 11.3
Tags: divisor , extremal , number theory
A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number.

2019 AIME Problems, 9
Tags: divisor
Call a positive integer $n$ $k$[i]-pretty[/i] if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$[i]-pretty[/i]. Let $S$ be the sum of positive integers less than $2019$ that are $20$[i]-pretty[/i]. Find $\tfrac{S}{20}$.

1992 Austrian-Polish Competition, 6
Tags: functional , functional equation , algebra , divisor , number theory
A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$) $f (1992 + x) = f (1992 - x)$ for all integers $x$. Can all positive divisors of $92$ occur as values of f?

2003 BAMO, 4
Tags: number theory , divisor , prime
An integer $n > 1$ has the following property: for every (positive) divisor $d$ of $n, d + 1$ is a divisor of $n + 1$. Prove that $n$ is prime.

2002 Kazakhstan National Olympiad, 7
Tags: prime divisor , divisor , divide , number theory
Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

1998 IMO, 3
Tags: number theoretic functions , prime factorization , divisor , number theory
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 $\frac{\tau (n^{2})}{\tau (n)}=m$.

2013 Austria Beginners' Competition, 1
Tags: divisor , number theory
Find all natural numbers $n> 1$ for which the following applies: The sum of the number $n$ and its second largest divisor is $2013$. (R. Henner, Vienna)

2016 Argentina National Olympiad, 5
Tags: divisor , algebra , prime divisor , number theory
Let $a$ and $b$ be rational numbers such that $a+b=a^2+b^2$ . Suppose the common value $s=a+b=a^2+b^2$ is not an integer, and let's write it as an irreducible fraction: $s=\frac{m}{n}$. Let $p$ be the smallest prime divisor of $n$. Find the minimum value of $p$.

2019 Junior Balkan Team Selection Tests - Romania, 1
Tags: divisor , number theory
For a positive integer $m$ we denote by $\tau (m)$ the number of its positive divisors, and by $\sigma (m)$ their sum. Determine all positive integers $n$ for which $n \sqrt{ \tau (n) }\le \sigma(n)$

2003 France Team Selection Test, 3
Tags: algebra , number theory , divisor , prime
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.

2025 Kyiv City MO Round 2, Problem 2
Tags: number theory , divisor
A positive integer \( n \) satisfies the following conditions: [list] [*] The number \( n \) has exactly \( 60 \) divisors: \( 1 = a_1 < a_2 < \cdots < a_{60} = n \); [*] The number \( n+1 \) also has exactly \( 60 \) divisors: \( 1 = b_1 < b_2 < \cdots < b_{60} = n+1 \). [/list] Let \( k \) be the number of indices \( i \) such that \( a_i < b_i \). Find all possible values of \( k \). [i]Note: Such numbers exist, for example, the numbers \( 4388175 \) and \( 4388176 \) both have \( 60 \) divisors.[/i] [i]Proposed by Anton Trygub[/i]

2018 Dutch IMO TST, 2
Tags: number theory , divisor
Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

2015 Dutch BxMO/EGMO TST, 1
Tags: number theory , divisor , divide
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

2010 Saudi Arabia Pre-TST, 3.2
Tags: perfect square , number theory , divisor
Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.

2018 Ukraine Team Selection Test, 7
Tags: perfect square , prime , divisor , number theory
The prime number $p > 2$ and the integer $n$ are given. Prove that the number $pn^2$ has no more than one divisor $d$ for which $n^2+d$ is the square of the natural number. .

2011 Rioplatense Mathematical Olympiad, Level 3, 6
Tags: number theory , algebra , divisor
Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$. Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.

2021 Dutch IMO TST, 4
Tags: number theory , divisor
Let $p > 10$ be prime. Prove that there are positive integers $m$ and $n$ with $m + n < p$ exist for which $p$ is a divisor of $5^m7^n-1$.

2020 Dürer Math Competition (First Round), P1
Tags: number theory , divisor
a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Is it possible that the product of all the positive divisors of two different natural numbers are equal?

2022 Switzerland - Final Round, 5
Tags: number theory , recurrence relation , divisor
For an integer $a \ge 2$, denote by $\delta_(a) $ the second largest divisor of $a$. Let $(a_n)_{n\ge 1}$ be a sequence of integers such that $a_1 \ge 2$ and $$a_{n+1} = a_n + \delta_(a_n)$$ for all $n \ge 1$. Prove that there exists a positive integer $k$ such that $a_k$ is divisible by $3^{2022}$.