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 Swedish Mathematical Competition, 5
Tags: number theory , divide
Show that $n$ divides $2^n + 1$ for infinitely many positive integers $n$.

1955 Kurschak Competition, 2
Tags: number theory , divide , digit
How many five digit numbers are divisible by $3$ and contain the digit $6$?

1988 Tournament Of Towns, (186) 3
Tags: number theory , sum , divide , divisible
Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

1998 Rioplatense Mathematical Olympiad, Level 3, 3
Tags: number theory , divide
Let $X$ be a finite set of positive integers. Prove that for every subset $A$ of $X$, there is a subset $B$ of $X$, with the following property: For each element $ e$ of $X$, $e$ divides an odd number of elements of $B$, if and only if $e$ is an element of $A$.

VMEO III 2006, 10.2
Tags: number theory , consecutive , dividisible , divide
Prove that among $39$ consecutive natural numbers, there is always a number that has sum of its digits divisible by $ 12$. Is it true if we replace $39$ with $38$?

1999 Singapore MO Open, 2
Tags: divisible , number theory , digit , divide
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

Mathley 2014-15, 8
Tags: number theory , divisible , power of 2 , divide
For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$ where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$. Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi

2020 Colombia National Olympiad, 4
Tags: arithmetic sequence , divide , number theory
Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$, then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.

2020 Malaysia IMONST 2, 3
Tags: divisible , number theory , divide
Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.

2009 Thailand Mathematical Olympiad, 10
Tags: divide , divisible , number theory
Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.

1995 Bulgaria National Olympiad, 1
Tags: divisible , number theory , divide
Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.

2005 Thailand Mathematical Olympiad, 12
Tags: divisible , number theory , divide , even
Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.

2006 QEDMO 2nd, 5
Tags: divide , number theory
For any natural number $m$, we denote by $\phi (m)$ the number of integers $k$ relatively prime to $m$ and satisfying $1 \le k \le m$. Determine all positive integers $n$ such that for every integer $k > n^2$, we have $n | \phi (nk + 1)$. (Daniel Harrer)

2008 China Northern MO, 3
Tags: number theory , prime factor , divide
Prove that: (1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$ (2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.

2011 Saudi Arabia Pre-TST, 4.4
Tags: number theory , divisor , divide
Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.

2009 Chile National Olympiad, 4
Tags: divisible , number theory , divide
Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$

1999 Tournament Of Towns, 1
Tags: divisible , divide , sum , combinatorics
For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order? (A Shapovalov)

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 $$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$

2013 Saudi Arabia Pre-TST, 1.2
Tags: divide , divisible , number theory
Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.

1970 Poland - Second Round, 3
Tags: divide , divisible , number theory
Prove the theorem: There is no natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $.

2004 Estonia National Olympiad, 4
Tags: divide , divisible , odd , number theory
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.

1989 Tournament Of Towns, (210) 4
Tags: divisible , number theory , divide
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 $k$ .

1954 Moscow Mathematical Olympiad, 267
Tags: divisible , algebra , divide , polynomial
Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$ then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.

1986 Austrian-Polish Competition, 7
Tags: factorial , divide , prime divisor
Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$. Prove that $k$ divides $n!$.

1979 Czech And Slovak Olympiad IIIA, 6
Tags: divide , prime , number theory
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.