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

2008 Bulgarian Autumn Math Competition, Problem 9.3
Tags: algebra , number theory , natural number , divisor , polynomial
Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

2016 All-Russian Olympiad, 3
Tags: number theory , greatest common divisor , divisor
Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N-2$.Find all possible values of $N$.

2012 Rioplatense Mathematical Olympiad, Level 3, 5
Tags: number theory , divisor
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.

2008 Thailand Mathematical Olympiad, 2
Tags: number theory , divisor
Find all positive integers $N$ with the following properties: (i) $N$ has at least two distinct prime factors, and (ii) if $d_1 < d_2 < d_3 < d_4$ are the four smallest divisors of $N$ then $N =d_1^2 + d_2 ^2+ d_3 ^2+ d_4^2$

2020 Final Mathematical Cup, 1
Tags: number theory , perfect square , divisor
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.

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

2002 Kazakhstan National Olympiad, 7
Tags: prime divisor , divisor , divide , number theory
Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

2020 Dürer Math Competition (First Round), P3
Tags: divisor , number theory
a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Show that if the product of all the positive divisors of two natural numbers are equal, then the two numbers must be equal.

2019 Germany Team Selection Test, 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.

2017 Bosnia and Herzegovina EGMO TST, 3
Tags: number theory , sequence , divisor
For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$

2024 Kyiv City MO Round 1, Problem 5
Tags: divisor , number theory
Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold: $$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$ [i]Proposed by Mykyta Kharin[/i]

2012 QEDMO 11th, 12
Tags: number theory , divisor , divide , divisible
Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.

2018 Grand Duchy of Lithuania, 4
Tags: divide , divisor , number theory
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 $n$.

2005 Germany Team Selection Test, 1
Tags: number theory , equation , 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 $n$.

2000 Czech and Slovak Match, 3
Tags: number theory , divisor , power of 2
Let $n$ be a positive integer. Prove that $n$ is a power of two if and only if there exists an integer $m$ such that $2^n-1$ is a divisor of $m^2 +9$.

2019 China Team Selection Test, 2
Tags: divisor , number theory
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 $$\sum_{d|n,d<n,d \in S} d = n$$

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

2013 Irish Math Olympiad, 8
Tags: number theory , diophantine equation , divisor
Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satis ed by at least two pairs of integers $(x, y)$ with $1 < x \le y$.

2007 Korea Junior Math Olympiad, 8
Tags: prime , positive integer , divisor , number theory
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$). Prove that there are infi nite number of [i]Primes of the Year[/i].

2012 Tournament of Towns, 2
Tags: number theory , divisor
The number $4$ has an odd number of odd positive divisors, namely $1$, and an even number of even positive divisors, namely $2$ and $4$. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?

2013 Balkan MO Shortlist, N4
Tags: divisible , number theory , 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 $p^2$.

2017 Ecuador Juniors, 5
Tags: number theory , consecutive , coprime , divisor
Two positive integers are coprime if their greatest common divisor is $1$. Let $C$ be the set of all divisors of the number $8775$ that are greater than $ 1$. A set of $k$ consecutive positive integers satisfies that each of them is coprime with some element of $C$. Determine the largest possible value of $K$.

1997 German National Olympiad, 2
Tags: inequalities , number theory , greatest odd divisor , divisor
For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$. Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.

2012 India Regional Mathematical Olympiad, 2
Tags: divisibility , divide , divisor , number theory
Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.

2020 German National Olympiad, 4
Tags: divisibility , divisor , number theory
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.