Found problems: 310
2017 China National Olympiad, 5
Let $D_n$ be the set of divisors of $n$. Find all natural $n$ such that it is possible to split $D_n$ into two disjoint sets $A$ and $G$, both containing at least three elements each, such that the elements in $A$ form an arithmetic progression while the elements in $G$ form a geometric progression.
2002 Mexico National Olympiad, 5
A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.
2020 Final Mathematical Cup, 1
Let $n$ be a given positive integer. Prove that there is no positive divisor $d$ of $2n^2$ such that $d^2n^2+d^3$ is a square of an integer.
2024 Belarusian National Olympiad, 10.1
Let $1=d_1<d_2<\ldots<d_k=n$ be all divisors of $n$. It turned out that numbers $d_2-d_1,\ldots,d_k-d_{k-1}$ are $1,3,\ldots,2k-3$ in some order. Find all possible values of $n$
[i]M. Zorka[/i]
2020 South Africa National Olympiad, 1
Find the smallest positive multiple of $20$ with exactly $20$ positive divisors.
2020 AIME Problems, 9
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m.$
The Golden Digits 2024, P1
Let $k\geqslant 2$ be a positive integer and $n>1$ be a composite integer. Let $d_1<\cdots<d_m$ be all the positive divisors of $n{}.$ Is it possible for $d_i+d_{i+1}$ to be a perfect $k$-th power, for every $1\leqslant i<m$?
[i]Proposed by Pavel Ciurea[/i]
2005 Germany Team Selection Test, 1
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 $n$.
1955 Moscow Mathematical Olympiad, 316
Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.
2014 Switzerland - Final Round, 9
The sequence of integers $a_1, a_2, ,,$ is defined as follows:
$$a_n=\begin{cases} 0\,\,\,\, if\,\,\,\, n\,\,\,\, has\,\,\,\, an\,\,\,\, even\,\,\,\, number\,\,\,\, of\,\,\,\, divisors\,\,\,\, greater\,\,\,\, than\,\,\,\, 2014 \\ 1 \,\,\,\, if \,\,\,\, n \,\,\,\, has \,\,\,\, an \,\,\,\, odd \,\,\,\, number \,\,\,\, of \,\,\,\, divisors \,\,\,\, greater \,\,\,\, than \,\,\,\, 2014\end{cases}$$
Show that the sequence $a_n$ never becomes periodic.
2012 Rioplatense Mathematical Olympiad, Level 3, 5
Let $a \ge 2$ and $n \ge 3$ be integers . Prove that one of the numbers $a^n+ 1 , a^{n + 1}+ 1 , ... , a^{2 n-2}+ 1$ does not share any odd divisor greater than $1$ with any of the other numbers.
2009 Estonia Team Selection Test, 6
For any positive integer $n$, let $c(n)$ be the largest divisor of $n$ not greater than $\sqrt{n}$ and let $s(n)$ be the least integer $x$ such that $n < x$ and the product $nx$ is divisible by an integer $y$ where $n < y < x$. Prove that, for every $n$, $s(n) = (c(n) + 1) \cdot \left( \frac{n}{c(n)}+1\right)$
2018 Dutch IMO TST, 2
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 $n$.
2018 Polish Junior MO Finals, 3
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.
2007 Indonesia TST, 2
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.
2022 Bulgarian Autumn Math Competition, Problem 10.3
Are there natural number(s) $n$, such that $3^n+1$ has a divisor in the form $24k+20$
2003 France Team Selection Test, 3
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.
2019 Brazil Team Selection Test, 1
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.
2013 Saudi Arabia IMO TST, 3
For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.
2013 NZMOC Camp Selection Problems, 12
For a positive integer $n$, let $p(n)$ denote the largest prime divisor of $n$. Show that there exist infinitely many positive integers m such that $p(m-1) < p(m) < p(m + 1)$.
2018 Danube Mathematical Competition, 1
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 $m$.
2017 Regional Competition For Advanced Students, 4
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]
2002 Junior Balkan Team Selection Tests - Romania, 1
Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers.
Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.
2018 Israel Olympic Revenge, 1
Let $n$ be a positive integer.
Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$
2019 Saudi Arabia IMO TST, 2
Find all pair of integers $(m,n)$ and $m \ge n$ such that there exist a positive integer $s$ and
a) Product of all divisor of $sm, sn$ are equal.
b) Number of divisors of $sm,sn$ are equal.