Olympiad problems

2019 Gulf Math Olympiad, 2

1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$. 2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$. 3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?

1972 Spain Mathematical Olympiad, 7

Tags: number theory , multiple , divide , divisible

Prove that for every positive integer $n$, the number $$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$ is a multiple of $8$.

Oliforum Contest V 2017, 1

Tags: number theory , digit , multiple

We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits? (Paolo Leonetti)

2010 Singapore Junior Math Olympiad, 2

Tags: multiple , number theory , digit

Find the sum of all the $5$-digit integers which are not multiples of $11$ and whose digits are $1, 3, 4, 7, 9$.

2003 May Olympiad, 3

Tags: number theory , divide , multiple

Find all pairs of positive integers $(a,b)$ such that $8b+1$ is a multiple of $a$ and $8a+1$ is a multiple of $b$.

2011 QEDMO 10th, 3

Tags: multiple , number theory , divide

Let $a, b$ be positive integers such that $a^2 + ab + 1$ a multiple of $b^2 + ab + 1$. Prove that $a = b$.

2009 Bundeswettbewerb Mathematik, 4

Tags: multiple , palindrome , number theory

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.

2018 NZMOC Camp Selection Problems, 1

Tags: multiple , divide , number theory

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 $12$.

2017 Irish Math Olympiad, 1

Tags: number theory , multiple , digit

Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$.

2012 NZMOC Camp Selection Problems, 4

Tags: number theory , multiple , divide , divisible , prime

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

2008 May Olympiad, 1

Tags: multiple , digit , number theory

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 $2008$?

2015 Puerto Rico Team Selection Test, 8

Tags: multiple , divisible , divide , number theory

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

1912 Eotvos Mathematical Competition, 2

Tags: number theory , multiple

Prove that for every positive integer $n$, the number $A_n = 5^n + 2 \cdot 3^{n-1} + 1$ is a multiple of $8$.

2019 Durer Math Competition Finals, 6

Tags: number theory , multiple , divide , divisible , digit

Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

2020 HK IMO Preliminary Selection Contest, 8

Tags: number theory , multiple

Find the smallest positive multiple of $77$ whose last four digits (from left to right) are $2020$.

2011 Belarus Team Selection Test, 1

Tags: combinatorics , number theory , multiple , divisible

Is it possible to arrange the numbers $1,2,...,2011$ over the circle in some order so that among any $25$ successive numbers at least $8$ numbers are multiplies of $5$ or $7$ (or both $5$ and $7$) ? I. Gorodnin

1988 Spain Mathematical Olympiad, 3

Tags: number theory , multiple

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.

2018 Singapore Junior Math Olympiad, 1

Tags: multiple , number theory , divide , digit

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 $37$.

2017 Saudi Arabia BMO TST, 3

Tags: number theory , multiple , digit

How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?

2011 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , digit , sum , multiple

a) Prove that if the sum of the non-zero digits $a_1, a_2, ... , a_n$ is a multiple of $27$, then it is possible to permute these digits in order to obtain an $n$-digit number that is a multiple of $27$. b) Prove that if the non-zero digits $a_1, a_2, ... , a_n$ have the property that every ndigit number obtained by permuting these digits is a multiple of $27$, then the sum of these digits is a multiple of $27$

2010 May Olympiad, 4

Tags: number theory , multiple

Find all natural numbers of $90$ digits that are multiples of $13$ and have the first $43$ digits equal to each other and nonzero, the last $43$ digits equal to each other, and the middle $4$ digits are $2, 0, 1, 0$, in that order.

2006 Korea Junior Math Olympiad, 1

Tags: algebra , number theory , combinatorics , multiple , product , permutation

$a_1, a_2,...,a_{2006}$ 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 $3$)

2014 Greece JBMO TST, 3

Tags: number theory , consecutive , sum , multiple

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 $i= 1,2,3, ... , (n-8)$

1977 Dutch Mathematical Olympiad, 3

Tags: number theory , divide , divisible , multiple

From each set $ \{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.

2007 Greece JBMO TST, 2

Tags: number theory , perfect square , multiple

Let $n$ be a positive integer such that $n(n+3)$ is a perfect square of an integer, prove that $n$ is not a multiple of $3$.