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

2002 Switzerland Team Selection Test, 3

Let $d_1,d_2,d_3,d_4$ be the four smallest divisors of a positive integer $n$ (having at least four divisors). Find all $n$ such that $d_1^2+d_2^2+d_3^2+d_4^2 = n$.

1999 All-Russian Olympiad Regional Round, 9.7

Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number $12$ has two prime factors: $2$ and $3$.)

2021 Bangladeshi National Mathematical Olympiad, 7

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 $100$.

2022 Greece National Olympiad, 2

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.

2024 Dutch IMO TST, 1

For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

2021 Durer Math Competition Finals, 5

Let $n$ be a positive integer. Show that every divisors of $2n^2 - 1$ gives a different remainder after division by $2n$.

2010 Saudi Arabia Pre-TST, 3.2

Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.

2005 AIME Problems, 3

Tags: divisor
How many positive integers have exactly three proper divisors, each of which is less than 50?

1997 Tuymaada Olympiad, 6

Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?

2012 Tournament of Towns, 4

Let $C(n)$ be the number of prime divisors of a positive integer $n$. (a) 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 $S$ finite or infinite? (b) Define $S'$ as a subset of S consisting of the pairs $(a, b)$ such that $C(a+b) > 1000$. Is $S'$ finite or infinite?

2013 IMO Shortlist, N7

Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called [i]good[/i] if \[a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.\] A good pair $(a,b)$ is called [i]excellent[/i] if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good. Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.

2013 IFYM, Sozopol, 3

The number $A$ is a product of $n$ distinct natural numbers. Prove that $A$ has at least $\frac{n(n-1)}{2}+1$ distinct divisors (including 1 and $A$).

2019 Germany 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.

2007 Peru MO (ONEM), 3

We say that a natural number of at least two digits $E$ is [i]special [/i] if each time two adjacent digits of $E$ are added, a divisor of $E$ is obtained. For example, $2124$ is special, since the numbers $2 + 1$, $1 + 2$ and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers such that they are all special.

2016 Dutch BxMO TST, 1

For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

2021 Thailand TSTST, 1

For each positive integer $n$, let $\rho(n)$ be the number of positive divisors of $n$ with exactly the same set of prime divisors as $n$. Show that, for any positive integer $m$, there exists a positive integer $n$ such that $\rho(202^n+1)\geq m.$

1995 Grosman Memorial Mathematical Olympiad, 1

Positive integers $d_1,d_2,...,d_n$ are divisors of $1995$. Prove that there exist $d_i$ and $d_j$ among them, such the denominator of the reduced fraction $d_i/d_j$ is at least $n$

2022 Saudi Arabia BMO + EGMO TST, 2.2

Find all positive integers $n$ that have precisely $\sqrt{n + 1}$ natural divisors.

2018 Singapore Junior Math Olympiad, 4

Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.

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 $n$, $s(n) = (c(n) + 1) \cdot \left( \frac{n}{c(n)}+1\right)$

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.

2016 NIMO Problems, 1

Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$. [i] Proposed by Michael Ren [/i]

2019 China Team Selection Test, 2

Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$ Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$

1999 Korea Junior Math Olympiad, 6

For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4>m$.

1985 Poland - Second Round, 2

Prove that for a natural number $ n > 2 $ the number $ n! $ is the sum of its $ n $ various divisors.