Found problems: 310
1995 All-Russian Olympiad Regional Round, 9.5
Find all prime numbers $p$ for which number $p^2 + 11$ has exactly six different divisors (counting $1$ and itself).
2022 China Team Selection Test, 5
Given a positive integer $n$, let $D$ is the set of positive divisors of $n$, and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent:
(a) For any positive divisor $m$ of $n$,
\[ n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}. \]
(b) For any positive divisor $k$ of $n$,
\[ k ~\Big|~ \sum_{d|k} f(d). \]
2019 AIME Problems, 9
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 $\tfrac{S}{20}$.
2017 Junior Balkan Team Selection Tests - Romania, 2
Let $n$ be a positive integer. For each of the numbers $1, 2,.., n$ we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least $0$ and at most
$n$.
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 $n$.
BIMO 2021, 1
Given a natural number $n$, call a divisor $d$ of $n$ to be $\textit{nontrivial}$ if $d>1$. A natural number $n$ is $\textit{good}$ if one or more distinct nontrivial divisors of $n$ sum up to $n-1$.
Prove that every natural number $n$ has a multiple that is good.
2013 Czech-Polish-Slovak Junior Match, 4
Determine the largest two-digit number $d$ with the following property:
for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$.
Note The numbers $a \ne 0, b$ and $c$ need not be different.
2009 Germany Team Selection Test, 2
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
2024 VJIMC, 4
Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.)
a) Prove that $(b_n)_{n \ge 0}$ is unbounded.
b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.
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.
Russian TST 2019, P1
A positive integer $n{}$ is called [i]discontinuous[/i] if for all its natural divisors $1 = d_0 < d_1 <\cdots<d_k$, written out in ascending order, there exists $1 \leqslant i \leqslant k$ such that $d_i > d_{i-1}+\cdots+d_1+d_0+1$. Prove that there are infinitely many positive integers $n{}$ such that $n,n+1,\ldots,n+2019$ are all discontinuous.
2024 Korea Junior Math Olympiad (First Round), 9.
Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares
2021 Dutch IMO TST, 4
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 Belarus Team Selection Test, 1
Let $S(n)$ be the sum of all different natural divisors of odd natural number $n> 1$ (including $n$ and $1$).
Prove that $(S(n))^3 <n^4$.
2018 Ukraine Team Selection Test, 7
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.
.
2023 Israel TST, P2
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 $B(n)$.
2011 Korea Junior Math Olympiad, 6
For a positive integer $n$, define the set $S_n$ as $S_n =\{(a, b)|a, b \in N, lcm[a, b] = n\}$ . Let $f(n)$ be the sum of $\phi (a)\phi (b)$ for all $(a, b) \in S_n$. If a prime $p$ relatively prime to $n$ is a divisor of $f(n)$, prove that there exists a prime $q|n$ such that $p|q^2 - 1$.
2017 South East Mathematical Olympiad, 4
For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 0, 1, 2, 3$.
Determine the smallest positive integer $m$ such that $f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017$.
2019 Hong Kong TST, 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.
2019 Regional Competition For Advanced Students, 4
Find all natural numbers $n$ that are smaller than $128^{97}$ and have exactly $2019$ divisors.
2019 Philippine TST, 4
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.
2023 IMO, 1
Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.
2011 Saudi Arabia Pre-TST, 4.4
Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.
1998 Switzerland Team Selection Test, 6
Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.
2019 Estonia Team Selection Test, 9
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.