Olympiad problems

2020 China Northern MO, P5

Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.

2022 APMO, 1

Tags: number theory , divisible , multiple

Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.

1995 Bundeswettbewerb Mathematik, 4

Tags: multiple , digit , number theory

Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.

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 $8$.

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

2015 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 $37$.

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$.

2024 Polish Junior MO Finals, 5

Tags: divisibility , multiple , digit , number theory

Let $S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}$. Show that the $2S$-digit number \[\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}\] is a multiple of $19$.

2011 Peru MO (ONEM), 1

Tags: number theory , multiple , consecutive

We say that a positive integer is [i]irregular [/i] if said number is not a multiple of none of its digits. For example, $203$ is irregular because $ 203$ is not a multiple of $2$, it is not multiple of $0$ and is not a multiple of $3$. Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$.

1910 Eotvos Mathematical Competition, 2

Tags: number theory , multiple , divide

Let $a, b, c, d$ and $u$ be integers such that each of the numbers $$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$ is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.

2018 Brazil National Olympiad, 5

Tags: number theory , sequence , multiple

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 $7$.

2010 Singapore Junior Math Olympiad, 2

Tags: digit , multiple , number theory

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

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?

2014 Singapore Junior Math Olympiad, 1

Tags: number theory , multiple , digit

Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.

2019 New Zealand MO, 4

Tags: number theory , divide , divisible , multiple

Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.

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 $b$.

1927 Eotvos Mathematical Competition, 1

Tags: number theory , multiple , divide

Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$ Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.

2011 QEDMO 9th, 4

Tags: factorial , number theory , multiple , divide

Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

2017 Grand Duchy of Lithuania, 4

Tags: factor , number theory , odd , multiple

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.)

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 $27$

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 $7$.

2003 May Olympiad, 3

Tags: number theory , multiple , sum of digits

Find the smallest positive integer that ends in $56$, is a multiple of $56$, and has the sum of its digits equal to $56$.

2022 New Zealand MO, 5

Tags: number theory , multiple , divide , divisible , recurrence relation

The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.

2013 Danube Mathematical Competition, 2

Tags: number theory , combinatorics , multiple , sum

Consider $64$ distinct natural numbers, at most equal to $2012$. Show that it is possible to choose four of them, denoted as $a,b,c,d$ such that $ a+b-c-d$ to be a multiple of $2013$

2019 Tournament Of Towns, 4

Tags: number theory , consecutive , multiple , sum of squares , circles

There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)