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

2011 Tournament of Towns, 1
Tags: divisor , number theory
An integer $N > 1$ is written on the board. Alex writes a sequence of positive integers, obtaining new integers in the following manner: he takes any divisor greater than $1$ of the last number and either adds it to, or subtracts it from the number itself. Is it always (for all $N > 1$) possible for Alex to write the number $2011$ at some point?

IV Soros Olympiad 1997 - 98 (Russia), 11.2
Tags: number theory , divisor
Find the three-digit number that has the greatest number of different divisors.

1982 Bundeswettbewerb Mathematik, 1
Tags: sum , number theory , divisor
Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.

1986 IMO Shortlist, 6
Tags: number theory , prime factorization , divisor
Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

2016 Danube Mathematical Olympiad, 2
Tags: number theory , prime number , divisor
Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime. [i]Lucian Petrescu[/i]

2019 Thailand 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.

2024 Romania EGMO TST, P4
Tags: number theory , divisor
Find all composite positive integers $a{}$ for which there exists a positive integer $b\geqslant a$ with the same number of divisors as $a{}$ with the following property: if $a_1<\cdots<a_n$ and $b_1<\cdots<b_n$ are the proper divisors of $a{}$ and $b{}$ respectively, then $a_i+b_i, 1\leqslant i\leqslant n$ are the proper divisors of some positive integer $c.{}$

2002 IMO Shortlist, 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.

2022 Greece National Olympiad, 2
Tags: number theory , divisor
Let $n>4$ be a positive integer, which is divisible by $4$. We denote by $A_n$ the sum of the odd positive divisors of $n$. We also denote $B_n$ the sum of the even positive divisors of $n$, excluding the number $n$ itself. Find the least possible value of the expression $$f(n)=B_n-2A_n,$$ for all possible values of $n$, as well as for which positive integers $n$ this minimum value is attained.

2013 Dutch Mathematical Olympiad, 4
Tags: number theory , equation , product , positive , divisor
For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$. For example, $P(20) = 8000$, as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$, whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$. (a) Find all positive integers $n$ satisfying $P(n) = 15n$. (b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$.

2021 New Zealand MO, 6
Tags: combinatorics , consecutive , divisor , number theory
Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied? $\bullet$ Each number (not in the top row) is a proper divisor of the number immediately above. $\bullet$ Each row consists of 1$0$ consecutive positive integers (but not necessarily in order).

2019 AMC 12/AHSME, 14
Tags: divisor
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

2018 Estonia Team Selection Test, 8
Tags: number theory , divisor
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 $d_2d_3 + d_3d_5 + d_5d_2 = n$.

2021 Bangladeshi National Mathematical Olympiad, 7
Tags: perfect square , number theory , function , perfect cube , divisor
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 $100$.

1992 Mexico National Olympiad, 4
Tags: number theory , divisor
Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$.

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine , diophantine equation , divisor
Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$. a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$. b) Find the number of all non-negative solutions of the above equation.

2023 Israel TST, P2
Tags: number theory , divisor
For each positive integer $n$, define $A(n)$ to be the sum of its divisors, and $B(n)$ to be the sum of products of pairs of its divisors. For example, \[A(10)=1+2+5+10=18\] \[B(10)=1\cdot 2+1\cdot 5+1\cdot 10+2\cdot 5+2\cdot 10+5\cdot 10=97\] Find all positive integers $n$ for which $A(n)$ divides $B(n)$.

2018 Switzerland - Final Round, 7
Tags: number theory , consecutive , divisor
Let $n$ be a natural integer and let $k$ be the number of ways to write $n$ as the sum of one or more consecutive natural integers. Prove that $k$ is equal to the number of odd positive divisors of $n$. Example: $9$ has three positive odd divisors and $9 = 9$, $9 = 4 + 5$, $9 = 2 + 3 + 4$.

2007 Korea Junior Math Olympiad, 8
Tags: prime , positive integer , divisor , number theory
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$). Prove that there are infi nite number of [i]Primes of the Year[/i].

2019 Hanoi Open Mathematics Competitions, 10
Tags: number theory , divisor , odd , sum
For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute $T =r_{100}+ r_{101} + r_{102}+...+r_{200}$

2016 AMC 10, 22
Tags: number theory , divisor
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have? $\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$

2024 Kyiv City MO Round 1, Problem 5
Tags: number theory , divisor
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]

2020 Malaysia IMONST 1, 17
Tags: divisor , number theory
Given a positive integer $n$. The number $2n$ has $28$ positive factors, while the number $3n$ has $30$ positive factors. Find the number of positive divisors of $6n$.

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 $5^m7^n-1$.

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