Found problems: 74
2004 Korea Junior Math Olympiad, 3
For an arbitrary prime number $p$, show that there exists infinitely many multiples of $p$ that can be expressed as the form $$\frac{x^2+y+1}{x+y^2+1}$$ Where $x, y$ are some positive integers.
2007 Portugal MO, 3
Determines the largest integer $n$ that is a multiple of all positive integers less than $\sqrt{n}$.
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 $xy^2 + 7$.
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.
2009 Bundeswettbewerb Mathematik, 4
A positive integer is called [i]decimal palindrome[/i] if its decimal representation $z_n...z_0$ with $z_n\ne 0$ is mirror symmetric, i.e. if $z_k = z_{n-k}$ applies to all $k= 0, ..., n$. Show that each integer that is not divisible by $10$ has a positive multiple, which is a decimal palindrome.
2000 Tuymaada Olympiad, 1
Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?
1972 Spain Mathematical Olympiad, 7
Prove that for every positive integer $n$, the number
$$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$
is a multiple of $8$.
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.
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 $i= 1,2,3, ... , (n-8)$
1999 Chile National Olympiad, 5
Consider the numbers $x_1, x_2,...,x_n$ that satisfy:
$\bullet$ $x_i \in \{-1,1\}$, with $i = 1, 2,...,n$
$\bullet$ $x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0$
Prove that $n$ is a multiple of $4$.
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 $\{a_n | n \in N*\}$.
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.
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?
2018 Auckland Mathematical Olympiad, 1
Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.
2018 NZMOC Camp Selection Problems, 1
Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$ must be a multiple of $12$.
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 $36$?
2000 Rioplatense Mathematical Olympiad, Level 3, 1
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.
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.
2017 Irish Math Olympiad, 1
Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$.
2011 QEDMO 10th, 3
Let $a, b$ be positive integers such that $a^2 + ab + 1$ a multiple of $b^2 + ab + 1$. Prove that $a = b$.
2006 Korea Junior Math Olympiad, 1
$a_1, a_2,...,a_{2006}$ is a permutation of $1,2,...,2006$.
Prove that $\prod_{i = 1}^{2006} (a_{i}^2-i) $ is a multiple of $3$. ($0$ is counted as a multiple of $3$)
1988 Spain Mathematical Olympiad, 3
Prove that if one of the numbers $25x+31y, 3x+7y$ (where $x,y \in Z$) is a multiple of $41$, then so is the other.
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.
2008 May Olympiad, 1
How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?
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 $9$)?