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

2024 Bulgarian Autumn Math Competition, 12.3
Tags: divisor , number theory
Let $n \geq 2$ be a positive integer. If $m$ is a positive integer, for which all of its positive divisors can be split into $n$ disjoint sets of equal sum, prove that $m \geq 2^{n+1}-2$

2016 Switzerland Team Selection Test, Problem 7
Tags: divisor , number theory
Find all positive integers $n$ such that $$\sum_{d|n, 1\leq d <n}d^2=5(n+1)$$

2017 Iberoamerican, 5
Tags: divisor , combinatorics
Given a positive integer $n$, all of its positive integer divisors are written on a board. Two players $A$ and $B$ play the following game: Each turn, each player colors one of these divisors either red or blue. They may choose whichever color they wish, but they may only color numbers that have not been colored before. The game ends once every number has been colored. $A$ wins if the product of all of the red numbers is a perfect square, or if no number has been colored red, $B$ wins otherwise. If $A$ goes first, determine who has a winning strategy for each $n$.

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 $3^{2022}$.

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 $d (n)$?

2016 Dutch IMO TST, 2
Tags: divisor , exponential , number theory , divisible
Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

2012 Peru MO (ONEM), 1
Tags: number theory , divisor
For each positive integer $n$ whose canonical decomposition is $n = p_1^{a_1} \cdot p_2^{a_2} \cdot\cdot\cdot p_k^{a_k}$, we define $t(n) = (p_1 + 1) \cdot (p_2 + 1) \cdot\cdot\cdot (p_k + 1)$. For example, $t(20) = t(2^2\cdot 5^1) = (2 + 1) (5 + 1) = 18$, $t(30) = t(2^1\cdot 3^1\cdot 5^1) = (2 + 1) (3 + 1) (5 + 1) = 72$ and $t(125) = t(5^3) = (5 + 1) = 6$ . We say that a positive integer $n$ is [i]special [/i]if $t(n)$ is a divisor of $n$. How many positive divisors of the number $54610$ are special?

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 $n$.

2017 Regional Competition For Advanced Students, 4
Tags: sum of squares , divisibility , divisor , number theory
Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$ where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. [i]Proposed by Walther Janous[/i]

2004 IMO Shortlist, 1
Tags: equation , number theory , divisor
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 $n$.

2020 South Africa National Olympiad, 1
Tags: number theory , divisor
Find the smallest positive multiple of $20$ with exactly $20$ positive divisors.

2024 Francophone Mathematical Olympiad, 4
Tags: number theory , prime number , divisor
Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.

Mathematical Minds 2024, P1
Tags: divisor , number theory
Find all positive integers $n\geqslant 2$ such that $d_{i+1}/d_i$ is an integer for all $1\leqslant i < k$, where $1=d_1<d_2<\dots <d_k=n$ are all the positive divisors of $n$. [i]Proposed by Pavel Ciurea[/i]

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 $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

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.

2016 Flanders Math Olympiad, 2
Tags: power , divide , divisor , number theory
Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.

2021 New Zealand MO, 6
Tags: number theory , divisor , consecutive , combinatorics
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).

2021 Dutch IMO TST, 4
Tags: divisor , number theory
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 $5^m7^n-1$.

2009 Argentina National Olympiad, 2
Tags: number theory , divisor
A positive integer $n$ is [i]acceptable [/i] if the sum of the squares of its proper divisors is equal to $2n+4$ (a divisor of $n$ is [i]proper [/i] if it is different from $1$ and of $n$ ). Find all acceptable numbers less than $10000$,

2012 Tournament of Towns, 2
Tags: number theory , divisor , prime
Let $C(n)$ be the number of prime divisors of a positive integer n. (For example, $C(10) = 2,C(11) = 1, C(12) = 2$). Consider set S of all pairs of positive integers $(a, b)$ such that $a\ne b$ and $C(a + b) = C(a) + C(b)$. Is set $S$ finite or infinite?

2019 Romania National Olympiad, 1
Tags: divisor , inequalities , number theory
Consider $A$, the set of natural numbers with exactly $2019$ natural divisors , and for each $n \in A$, denote $$S_n=\frac{1}{d_1+\sqrt{n}}+\frac{1}{d_2+\sqrt{n}}+...+\frac{1}{d_{2019}+\sqrt{n}}$$ where $d_1,d_2, .., d_{2019}$ are the natural divisors of $n$. Determine the maximum value of $S_n$ when $n$ goes through the set $ A$.

2014 EGMO, 3
Tags: quadratic , combinatorial number theory , divisor , number theory
We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

2001 All-Russian Olympiad Regional Round, 9.6
Tags: last digit , digit , number theory , divisor
Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?

2011 Cuba MO, 3
Tags: divisor , number theory
Let $n$ be a positive integer and let $$1 = d_1 < d_2 < d_3 < d_4$$ the four smallest divisors of $n$. Find all$ n$ such that $$n^2 = d_1 + d_2^2+d_3^3 +d_4^4.$$

2014 Contests, 2
Tags: number theory , divisor
Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.