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

2019 Hong Kong 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 Korea Junior Math Olympiad (First Round), 9.
Tags: number theory , divisor
Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares

2013 Balkan MO Shortlist, N4
Tags: number theory , divisible , sum of powers , divisor
Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$.

2002 BAMO, 4
Tags: inequalities , number theory , divisor
For $n \ge 1$, let $a_n$ be the largest odd divisor of $n$, and let $b_n = a_1+a_2+...+a_n$. Prove that $b_n \ge \frac{ n^2 + 2}{3}$, and determine for which $n$ equality holds. For example, $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3$, thus $b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23$ .

1992 ITAMO, 3
Tags: factorial , sum of divisors , divisor , number theory
Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

1989 Bundeswettbewerb Mathematik, 4
Tags: egyptian fractions , number theory , divisor
Let $n$ be an odd positive integer. Show that the equation $$ \frac{4}{n} =\frac{1}{x} + \frac{1}{y}$$ has a solution in the positive integers if and only if $n$ has a divisor of the form $4k+3$.

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 $d_4+d_5=d_6+7$.

2014 Czech-Polish-Slovak Junior Match, 5
Tags: game , number theory , divide , divisor
There is the number $1$ on the board at the beginning. If the number $a$ is written on the board, then we can also write a natural number $b$ such that $a + b + 1$ is a divisor of $a^2 + b^2 + 1$. Can any positive integer appear on the board after a certain time? Justify your answer.

2022 IFYM, Sozopol, 7
Tags: number theory , divisor
Let’s note the set of all integers $n>1$ which are not divisible by a square of a prime number. We define the number $f(n)$ as the greatest amount of divisors of $n$ which could be chosen in such way so that for each two chosen $a$ and $b$, not necessarily different, the number $a^2+ab+b^2+n$ is not a square. Find all $m$ for which there exists $n$ so that $f(n)=m$.

2016 Saudi Arabia GMO TST, 3
Tags: divide , divisor , algebra , polynomial
Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions: $x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$

2009 Bundeswettbewerb Mathematik, 2
Tags: number theory , quadratic trinomial , trinomial , prime , divisor , prime divisor
Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

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 $T =r_{100}+ r_{101} + r_{102}+...+r_{200}$

2018 Turkey Junior National Olympiad, 1
Tags: number theory , divisor
Let $s(n)$ be the number of positive integer divisors of $n$. Find the all positive values of $k$ that is providing $k=s(a)=s(b)=s(2a+3b)$.

2014 IFYM, Sozopol, 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 $d_4+d_5=d_6+7$.

2015 Czech-Polish-Slovak Junior Match, 5
Tags: divisor , number theory
Determine all natural numbers$ n> 1$ with the property: For each divisor $d> 1$ of number $n$, then $d - 1$ is a divisor of $n - 1$.

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 $d_2d_3 + d_3d_5 + d_5d_2 = n$.

2025 AIME, 7
Tags: probability , combinatorics , divisor
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Russian TST 2019, P1
Tags: divisor , number theory
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.

1983 IMO Shortlist, 2
Tags: number theory , sum of divisors , inequality , divisor
Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

2001 All-Russian Olympiad Regional Round, 9.6
Tags: number theory , last digit , digit , 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?

2000 Portugal MO, 3
Tags: number theory , divide , divisor
Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

2015 Dutch IMO TST, 5
Tags: number theory , divisor , maximum , exponential
For a positive integer $n$, we de ne $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$. 1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer. 2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.

2015 Azerbaijan National Olympiad, 4
Tags: prime divisor , divisor , number theory
Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.

2016 India Regional Mathematical Olympiad, 3
Tags: number theory , divide , divisor
$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$

1981 Austrian-Polish Competition, 7
Tags: number theory , divisor , prime divisor
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.