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$.

2022 New Zealand MO, 5

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

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 $n$.

1994 Greece National Olympiad, 4

Tags: multiple , number theory

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

1977 Dutch Mathematical Olympiad, 3

Tags: divisible , number theory , divide , 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 $7$.

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 $2$?

2003 May Olympiad, 1

Tags: number theory , multiple

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 $36$?

2018 NZMOC Camp Selection Problems, 1

Tags: multiple , number theory , divide

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 $12$.

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

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 $2$.

2000 Tuymaada Olympiad, 1

Tags: number theory , digit , multiple , maximum

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?

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 $167$.

2007 Korea Junior Math Olympiad, 1

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

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 $5$.

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.

1972 Spain Mathematical Olympiad, 7

Tags: divisible , number theory , multiple , divide

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

2013 IMAC Arhimede, 1

Tags: divisible , multiple , number theory

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 $10$.

2009 Korea Junior Math Olympiad, 1

Tags: prime , multiple , product , number theory

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 $c$ $\bullet$ $b + c = a^2 - 1$.

2008 May Olympiad, 1

Tags: multiple , number theory , digit

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 $2008$?

1997 Moldova Team Selection Test, 3

Tags: number theory , pigeonhole principle , 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.

2003 Korea Junior Math Olympiad, 4

Tags: multiple , integer , combinatorics

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.

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 $169$.

2015 Puerto Rico Team Selection Test, 8

Tags: divisible , number theory , multiple , divide

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 $5$.

2009 Bundeswettbewerb Mathematik, 4

Tags: number theory , palindrome , multiple

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.

2011 QEDMO 10th, 3

Tags: number theory , multiple , divide

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

2011 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , sum , multiple , digit

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 $27$