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: 408

2018 Saudi Arabia GMO TST, 2
Tags: digit , divisible , number theory
Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.

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 $k = 15$.

2011 Hanoi Open Mathematics Competitions, 1
Tags: divisible , number theory
An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$. How many integers between $1$ and $100$ are octal? (A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$

2015 Caucasus Mathematical Olympiad, 2
Tags: maximum , number theory , divisible
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?

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 $7$.

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}] $.

2019 Serbia JBMO TST, 1
Tags: number theory , divisible , factorial
Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?

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 $3$.

1997 Tournament Of Towns, (524) 1
Tags: sum of digits , divisible , number theory
How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$? (AI Galochkin)

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

2017 Junior Regional Olympiad - FBH, 4
Tags: statements , prime , divisible
Let $n$ and $k$ be positive integers for which we have $4$ statements: $i)$ $n+1$ is divisible with $k$ $ii)$ $n=2k+5$ $iii)$ $n+k$ is divisible with $3$ $iv)$ $n+7k$ is prime Determine all possible values for $n$ and $k$, if out of the $4$ statements, three of them are true and one is false

2011 Regional Olympiad of Mexico Center Zone, 4
Tags: digit , divisible , number theory
Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.

1987 All Soviet Union Mathematical Olympiad, 444
Tags: divisible , number theory
Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.

2018 Czech-Polish-Slovak Junior Match, 4
Tags: digit , divisible , number theory
Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.

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

1953 Moscow Mathematical Olympiad, 250
Tags: number theory , combinatorics , divisible
Somebody wrote $1953$ digits on a circle. The $1953$-digit number obtained by reading these figures clockwise, beginning at a certain point, is divisible by $27$. Prove that if one begins reading the figures at any other place, one gets another $1953$-digit number also divisible by $27$.

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 $n^3 - 1$.

2015 Belarus Team Selection Test, 1
Tags: divisible , number theory , consecutive , prime
Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA

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 $2^n$.