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.

AND:
OR:
NO:

Found problems: 408

1989 Poland - Second Round, 4
Tags: number theory , divide , divisible
The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.

2015 JBMO Shortlist, NT2
Tags: divisible , number theory , repunit
A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$ . Prove that: α) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$ b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$

1993 Swedish Mathematical Competition, 1
Tags: number theory , sum of digits , divisible , divide
An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.

2022 New Zealand MO, 6
Tags: number theory , divisible , sum
Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.

1979 Chisinau City MO, 174
Tags: number theory , odd , divide , divisible
Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.

2013 IMAC Arhimede, 1
Tags: divisible , multiple , number theory
Show that in any set of three distinct integers there are two of them, say $a$ and $b$ such that the number $a^5b^3-a^3b^5$ is a multiple of $10$.

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.

2021 New Zealand MO, 3
Tags: number theory , divisible
In a sequence of numbers, a term is called [i]golden [/i] if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, . . . , 2021$?

2014 Switzerland - Final Round, 5
Tags: number theory , divisible , divide
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 $a_1 + a_2 + a_3 + a_6 = 2^6.$

2018 India PRMO, 3
Tags: divisible , number theory , digit
Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.

1938 Moscow Mathematical Olympiad, 042
Tags: number theory , divisible , combinatorics
How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?

1949 Moscow Mathematical Olympiad, 168
Tags: sum , divisible , number theory
Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.

1997 Chile National Olympiad, 2
Tags: number theory , divisible , divide
Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$. Prove that $a \equiv n \equiv 1 \mod 10$ .

1981 Spain Mathematical Olympiad, 8
Tags: sum of squares , divisible , number theory
If $a$ is an odd number, show that $$a^4 + 4a^3 + 11a^2 + 6a+ 2$$ is a sum of three squares and is divisible by $4$.

2015 Puerto Rico Team Selection Test, 3
Tags: number theory , divisible , trinomial , quadratic
Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.

1997 Singapore MO Open, 2
Tags: number theory , binomial , divide , divisible
Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.

2014 Bosnia and Herzegovina Junior BMO TST, 1
Tags: digit , divisible , number theory
Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.

2003 Austrian-Polish Competition, 7
Tags: number theory , divide , factorial , divisible
Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $. Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .

1990 Greece National Olympiad, 3
Tags: number theory , divide , divisible
For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

2024 Regional Competition For Advanced Students, 4
Tags: number theory , divisible , divide
Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

1995 Czech And Slovak Olympiad IIIA, 4
Tags: digit , divisible , number theory
Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?

2021 Polish Junior MO Second Round, 3
Tags: number theory , divisible , divide
Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.

1970 Polish MO Finals, 3
Tags: number theory , divisible , prime , binomial coefficient
Prove that an integer $n > 1$ is a prime number if and only if, for every integer $k$ with $1\le k \le n-1$, the binomial coefficient $n \choose k$ is divisible by $n$.

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.

1970 All Soviet Union Mathematical Olympiad, 137
Tags: combinatorics , number theory , divisible , sum
Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.