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

1975 Chisinau City MO, 102
Tags: divide , digit , number theory , divisible , combinatorics
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 $k = 15$.

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

1960 Poland - Second Round, 4
Tags: divide , divisible , number theory
Prove that if $ n $ is a non-negative integer, then number $$ 2^{n+2} + 3^{2n+1}$$ is divisible by $7$.

2020 Durer Math Competition Finals, 6
Tags: divide , number theory
Positive integers $a, b$ and $c$ are all less than $2020$. We know that $a$ divides $b + c$, $b$ divides $a + c$ and $c$ divides $a + b$. How many such ordered triples $(a, b, c)$ are there? Note: In an ordered triple, the order of the numbers matters, so the ordered triple $(0, 1, 2)$ is not the same as the ordered triple $(2, 0, 1)$.

1976 Spain Mathematical Olympiad, 4
Tags: divide , number theory
Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

2004 Thailand Mathematical Olympiad, 15
Tags: divide , divisible , number theory
Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.

2003 Estonia National Olympiad, 4
Tags: integer , divide , infinity , divisible , number theory , floor function
Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.

2022 Federal Competition For Advanced Students, P2, 1
Tags: divide , number theory , function
Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

2022 Switzerland - Final Round, 2
Tags: remainder , divide , number theory
Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$.

1998 Tournament Of Towns, 3
Tags: divide , divisible , combinatorics
Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.) (G Galperin)

2019 New Zealand MO, 4
Tags: divide , divisible , number theory , multiple
Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.

1974 Dutch Mathematical Olympiad, 2
Tags: divide , divisible , number theory
$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

2021 Saudi Arabia BMO TST, 3
Tags: divide , number theory , divisible
Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.

2015 Saudi Arabia BMO TST, 4
Tags: divide , divisible , number theory
Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$. Lê Anh Vinh

1994 Austrian-Polish Competition, 7
Tags: divide , divisible , number theory
Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.

2011 Belarus Team Selection Test, 2
Tags: divide , number theory , divisible
Do they exist natural numbers $m,x,y$ such that $$m^2 +25 \vdots (2011^x-1007^y) ?$$ S. Finskii

2013 Abels Math Contest (Norwegian MO) Final, 3
Tags: divide , prime , number theory
A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

1955 Moscow Mathematical Olympiad, 292
Tags: divide , number theory
Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$. Prove that if $n > 3$, then $6$ divides $ab$.

2016 May Olympiad, 3
Tags: digit , divide , number theory
We say that a positive integer is [i]quad-divi[/i] if it is divisible by the sum of the squares of its digits, and also none of its digits is equal to zero. a) Find a quad-divi number such that the sum of its digits is $24$. b) Find a quad-divi number such that the sum of its digits is $1001$.

2010 Saudi Arabia IMO TST, 1
Tags: divide , divisible , number theory
Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.

2010 Saudi Arabia IMO TST, 3
Tags: divide , number theory , recurrence relation
Consider the sequence $a_1 = 3$ and $a_{n + 1} =\frac{3a_n^2+1}{2}-a_n$ for $n = 1 ,2 ,...$. Prove that if $n$ is a power of $3$ then $n$ divides $a_n$ .

2000 Chile National Olympiad, 5
Tags: number theory , divide , power of 2 , divisible
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 $2^n$.

2007 Cuba MO, 5
Tags: digit , divide , power of 2 , number theory
Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.

2017 QEDMO 15th, 10
Tags: divide , divisor , number theory
Let $p> 3$ be a prime number and let $q = \frac{4^p-1}{3}$. Show that $q$ is a composite integer as well is a divisor of $2^{q-1}- 1$.

2021 Dutch IMO TST, 3
Tags: divide , divisible , number theory
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.