Olympiad problems

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$.

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

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 $37$.

2009 Korea Junior Math Olympiad, 1

Tags: number theory , multiple , product , prime

For primes $a, b,c$ that satisfy the following, calculate $abc$. $\bullet$ $b + 8$ is a multiple of $a$, $\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$ $\bullet$ $b + c = a^2 - 1$.

2023 Brazil Team Selection Test, 4

Tags: number theory , prime number , multiple

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.

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 $2020$.

2020 HK IMO Preliminary Selection Contest, 15

Tags: number theory , multiple

How many ten-digit positive integers consist of ten different digits and are divisible by $99$?

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.

2014 May Olympiad, 3

Tags: combinatorics , game , number theory , multiple

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.

1993 Spain Mathematical Olympiad, 4

Tags: number theory , prime , multiple

Prove that for each prime number distinct from $2$ and $5$ there exist infinitely many multiples of $p$ of the form $1111...1$.

2004 Junior Balkan Team Selection Tests - Moldova, 5

Tags: number theory , multiple , power of 5

The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$.

2010 May Olympiad, 1

Tags: number theory , multiple

Determine the smallest positive integer that has all its digits equal to $4$, and is a multiple of $169$.

2021 Puerto Rico Team Selection Test, 4

Tags: digit , multiple , divisible , number theory

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 $30$?

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 $3$.

1990 IMO Shortlist, 20

Tags: pigeonhole principle , number theory , decimal representation , divisibility , multiple , digit

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.

2007 QEDMO 4th, 11

Tags: number theory , multiple , set

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 $m.$

2007 Korea Junior Math Olympiad, 1

Tags: number theory , number theory with sequences , integer sequence , multiple , sequence

A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satisfies 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 $5$.

2001 Chile National Olympiad, 4

Tags: number theory , multiple , divisible

Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.

2013 IMAC Arhimede, 1

Tags: number theory , multiple , divisible

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 $10$.

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 $5$.

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 $30$?

2019 Gulf Math Olympiad, 2

Tags: number theory , digit , multiple

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 $2$?

1994 Greece National Olympiad, 4

Tags: number theory , multiple

How many sums $$x_1+x_2+x_3, \ \ 1\leq x_j\leq 300, \ j=1,2,3$$ are multiples of $3$;

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)