Found problems: 74
2007 Korea Junior Math Olympiad, 1
A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satis fies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.
2003 May Olympiad, 1
Four digits $a, b, c, d$, different from each other and different from zero, are chosen and the list of all the four-digit numbers that are obtained by exchanging the digits $a, b, c, d$ is written. What digits must be chosen so that the list has the greatest possible number of four-digit numbers that are multiples of $36$?
2017 Puerto Rico Team Selection Test, 3
Given are $n$ integers. Prove that at least one of the following conditions applies:
1) One of the numbers is a multiple of $n$.
2) You can choose $k\le n$ numbers whose sum is a multiple of $ n$.
1999 Cono Sur Olympiad, 4
Let $A$ be a six-digit number, three of which are colored and equal to $1, 2$, and $4$.
Prove that it is always possible to obtain a number that is a multiple of $7$, by performing only one of the following operations: either delete the three colored figures, or write all the numbers of $A$ in some order.
2014 Korea Junior Math Olympiad, 5
For positive integers $x,y$, find all pairs $(x,y)$ such that $x^2y + x$ is a multiple of $xy^2 + 7$.
2009 Junior Balkan Team Selection Tests - Romania, 1
For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times.
Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.
2014 Contests, 3
Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ .
Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ .
Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$
2015 Singapore Junior Math Olympiad, 1
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.
2001 Chile National Olympiad, 4
Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.
2009 Grand Duchy of Lithuania, 1
The natural number $N$ is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$.
(a) Find one such number.
(b) Find the smallest such number.
2007 QEDMO 4th, 11
Let $S_{1},$ $S_{2},$ $...,$ $S_{n}$ be finitely many subsets of $\mathbb{N}$ such that $S_{1}\cup S_{2}\cup...\cup S_{n}=\mathbb{N}.$ Prove that there exists some $k\in\left\{ 1,2,...,n\right\} $ such that for each positive integer $m,$ the set $S_{k}$ contains infinitely many multiples of $m.$
2019 New Zealand MO, 4
Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.
2023 Brazil Team Selection Test, 4
Find all positive integers $n$ with the following property: There are only a finite number of positive multiples of $n$ that have exactly $n$ positive divisors.
2002 Nordic, 4
Eva, Per and Anna play with their pocket calculators. They choose different integers and check, whether or not they are divisible by ${11}$. They only look at nine-digit numbers consisting of all the digits ${1, 2, . . . , 9}$. Anna claims that the probability of such a number to be a multiple of ${11}$ is exactly ${1/11}$. Eva has a different opinion: she thinks the probability is less than ${1/11}$. Per thinks the probability is more than ${1/11}$. Who is correct?
2014 May Olympiad, 3
Ana and Luca play the following game. Ana writes a list of $n$ different integer numbers. Luca wins if he can choose four different numbers, $a, b, c$ and $d$, so that the number $a+b-(c+d)$ is multiple of $20$. Determine the minimum value of $n$ for which, whatever Ana's list, Luca can win.
1990 IMO Longlists, 65
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
1990 IMO Shortlist, 20
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
2017 Grand Duchy of Lithuania, 4
Show that there are infinitely many positive integers $n$ such that the number of distinct odd prime factors of $n(n + 3)$ is a multiple of $3$.
(For instance, $180 = 2^2 \cdot 3^2 \cdot 5$ has two distinct odd prime factors and $840 = 2^3 \cdot 3 \cdot 5 \cdot 7$ has three.)
2003 Korea Junior Math Olympiad, 4
When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.
2018 Brazil National Olympiad, 5
Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.
1997 Moldova Team Selection Test, 3
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
2021 Puerto Rico Team Selection Test, 4
How many numbers $\overline{abcd}$ with different digits satisfy the following property:
if we replace the largest digit with the digit $1$ results in a multiple of $30$?
I Soros Olympiad 1994-95 (Rus + Ukr), 9.10
For which natural $n$ there exists a natural number multiple of $n$, whose decimal notation consists only of the digits $8$ and $9$ (possibly only from numbers $8$ or only from numbers $9$)?
2009 Korea Junior Math Olympiad, 1
For primes $a, b,c$ that satis fy the following, calculate $abc$.
$\bullet$ $b + 8$ is a multiple of $a$,
$\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$
$\bullet$ $b + c = a^2 - 1$.
2010 May Olympiad, 1
Determine the smallest positive integer that has all its digits equal to $4$, and is a multiple of $169$.