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: 387

2018 Stanford Mathematics Tournament, 1
Tags: number theory , divide , divisible
Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.

2013 Cuba MO, 8
Tags: divide , number theory
Prove that there are infinitely many pairs $(a, b)$ of positive integers with the following properties: $\bullet$ $a+b$ divides $ab+1$, $\bullet$ $a-b$ divides $ab -1$, $\bullet$ $b > 2$ and $a > b\sqrt3 - 1$.

1990 Tournament Of Towns, (265) 3
Tags: number theory , divide , sum
Find $10$ different positive integers such that each of them is a divisor of their sum (S Fomin, Leningrad)

2019 Saudi Arabia Pre-TST + Training Tests, 1.2
Tags: number theory , divide , arithmetic sequence
Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .

2015 Saudi Arabia IMO TST, 1
Tags: number theory , divide , divisible , gcd
Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$. Trần Nam Dũng

1998 Singapore Senior Math Olympiad, 1
Tags: integer , divisible , divide , number theory
Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

2003 Bosnia and Herzegovina Junior BMO TST, 3
Tags: divide , divisible , number theory
Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.

2018 Bosnia and Herzegovina EGMO TST, 2
Tags: positive integer , divide , number theory
Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds $$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$

2008 Postal Coaching, 3
Tags: number theory , divide , divisible
Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.

2016 Grand Duchy of Lithuania, 4
Tags: divide , divisible , number theory
Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.

1989 Tournament Of Towns, (210) 4
Tags: number theory , divide , divisible
Prove that if $k$ is an even positive integer then it is possible to write the integers from $1$ to $k-1$ in such an order that the sum of no set of successive numbers is divisible by $k$ .

1965 German National Olympiad, 2
Tags: number theory , divide
Determine which of the prime numbers $2,3,5,7,11,13,109,151,491$ divide $z =1963^{1965} -1963$.

2014 Switzerland - Final Round, 5
Tags: number theory , divide , divisible
Let $a_1, a_2, ...$ a sequence of integers such that for every $n \in N$ we have: $$\sum_{d | n} a_d = 2^n.$$ Show for every $n \in N$ that $n$ divides $a_n$. Remark: For $n = 6$ the equation is $a_1 + a_2 + a_3 + a_6 = 2^6.$

2003 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , consecutive , divide , sum of powers , prime
Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.

1995 Singapore MO Open, 4
Tags: number theory , divisible , divide
Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.

1972 Spain Mathematical Olympiad, 7
Tags: number theory , multiple , divide , divisible
Prove that for every positive integer $n$, the number $$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$ is a multiple of $8$.

1979 Czech And Slovak Olympiad IIIA, 6
Tags: number theory , divide , prime
Find all natural numbers $n$, $n < 10^7$, for which: If natural number $m$, $1 < m < n$, is not divisible by $n$, then $m$ is prime.

1975 Chisinau City MO, 102
Tags: combinatorics , number theory , divisible , divide , digit
Two people write a $2k$-digit number, using only the numbers $1, 2, 3, 4$ and $5$. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by $9$, if the first seeks to interfere with it? Consider the cases $k = 10$ and $k = 15$.

2019 Denmark MO - Mohr Contest, 3
Tags: number theory , divide
Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?

2018 Costa Rica - Final Round, 5
Tags: number theory , divisible , divide
Let $a$ and $ b$ be even numbers, such that $M = (a + b)^2-ab$ is a multiple of $5$. Consider the following statements: I) The unit digits of $a^3$ and $b^3$ are different. II) $M$ is divisible by $100$. Please indicate which of the above statements are true with certainty.

2005 Greece JBMO TST, 4
Tags: number theory , divide , factorial
Find all the positive integers $n , n\ge 3$ such that $n\mid (n-2)!$

2019 Durer Math Competition Finals, 5
Tags: divisible , divide , number theory
We want to write down as many distinct positive integers as possible, so that no two numbers on our list have a sum or a difference divisible by $2019$. At most how many integers can appear on such a list?

2017 Saudi Arabia IMO TST, 3
Tags: number theory , divide , divisible
Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.

2014 Regional Olympiad of Mexico Center Zone, 1
Tags: number theory , divisible , divide , product
Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

2000 Chile National Olympiad, 5
Tags: power of 2 , divide , divisible , number theory
Let $n$ be a positive number. Prove that there exists an integer $N =\overline{m_1m_2...m_n}$ with $m_i \in \{1, 2\}$ which is divisible by $2^n$.