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

2016 Saint Petersburg Mathematical Olympiad, 1
Tags: number theory , sequence , divisor , number theory with sequences , integer sequence
In the sequence of integers $(a_n)$, the sum $a_m + a_n$ is divided by $m + n$ with any different $m$ and $n$. Prove that $a_n$ is a multiple of $n$ for any $n$.

IV Soros Olympiad 1997 - 98 (Russia), 11.2
Tags: number theory , divisor
Find the three-digit number that has the greatest number of different divisors.

2021 Durer Math Competition Finals, 5
Tags: number theory , remainder , divisor
Let $n$ be a positive integer. Show that every divisors of $2n^2 - 1$ gives a different remainder after division by $2n$.

1982 Tournament Of Towns, (015) 1
Tags: number theory , divisible , divisor
Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors. (M Levin)

2002 Switzerland Team Selection Test, 3
Tags: number theory , sum , divisor
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 $d_1^2+d_2^2+d_3^2+d_4^2 = n$.

2019 Tournament Of Towns, 1
Tags: factor , inequalities , number theory , prime , number of divisors , prime factorization , divisor
Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity a) not greater than the complexity of $n$. b) less than the complexity of $n$. (Boris Frenkin)

1999 Korea Junior Math Olympiad, 6
Tags: divisor , number theory
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 $p(m)^4>m$.

2013 Regional Competition For Advanced Students, 1
Tags: probability , number theory , divisor
For which integers between $2000$ and $2010$ (including) is the probability that a random divisor is smaller or equal $45$ the largest?

2005 India IMO Training Camp, 2
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 $n$.

2015 European Mathematical Cup, 3
Tags: number theory , divisor
Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers. [i]Paulius Ašvydis[/i]

2019 AMC 12/AHSME, 14
Tags: divisor
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

2009 Flanders Math Olympiad, 2
Tags: number theory , divisor
A natural number has four natural divisors: $1$, the number itself, and two real divisors. That number plus $9$ is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.

2021 Dutch Mathematical Olympiad, 5
Tags: number theory , prime , divisor
We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.

2017 Junior Balkan Team Selection Tests - Romania, 2
Tags: inequalities , number theory , difference , divisor
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 $n$.

1998 IMO Shortlist, 6
Tags: number theoretic functions , prime factorization , divisor , number theory
For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

2013 IFYM, Sozopol, 3
Tags: number theory , divisor
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 $A$).

1992 Austrian-Polish Competition, 1
Tags: sum , positive , number theory , product , even , divisor
For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

2019 239 Open Mathematical Olympiad, 2
Tags: number theory , number of divisors , divisor
Is it true that there are $130$ consecutive natural numbers, such that each of them has exactly $900$ natural divisors?

2017 China Team Selection Test, 1
Tags: number theory , modular arithmetic , divisor
Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.

2021 New Zealand MO, 6
Tags: combinatorics , consecutive , divisor , number theory
Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied? $\bullet$ Each number (not in the top row) is a proper divisor of the number immediately above. $\bullet$ Each row consists of 1$0$ consecutive positive integers (but not necessarily in order).

1987 Mexico National Olympiad, 6
Tags: number theory , divisor
Prove that for every positive integer n the number $(n^3 -n)(5^{8n+4} +3^{4n+2})$ is a multiple of $3804$.

2002 IMO Shortlist, 3
Tags: algebra , number theory , divisor , prime
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.

2016 Dutch IMO TST, 2
Tags: number theory , divisible , divisor , exponential
Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

1986 IMO Longlists, 58
Tags: number theory , prime factorization , divisor
Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

2014 Federal Competition For Advanced Students, P2, 4
Tags: number theory , sum of squares , product , divisor , consecutive
For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?