Olympiad problems

2020 Switzerland Team Selection Test, 2

Find all positive integers $n$ such that there exists an infinite set $A$ of positive integers with the following property: For all pairwise distinct numbers $a_1, a_2, \ldots , a_n \in A$, the numbers $$a_1 + a_2 + \ldots + a_n \text{ and } a_1\cdot a_2\cdot \ldots\cdot a_n$$ are coprime.

2015 Saudi Arabia BMO TST, 4

Tags: number theory , coprime , gcd , prime

Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$. Malik Talbi

2015 IFYM, Sozopol, 6

Tags: coprime , number theory

The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers?

2006 Korea Junior Math Olympiad, 5

Tags: number theory , coprime , integer , positive integer

Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

1989 Romania Team Selection Test, 2

Tags: system of equations , algebra , number theory , coprime

Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

1969 IMO Shortlist, 18

Tags: number theory , coprime , frobenius , additive number theory

$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$

1997 Israel National Olympiad, 5

Tags: coprime , prime , number theory

The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime. Show that at least one of these numbers is prime.

2022 Bulgaria JBMO TST, 3

Tags: number theory , coprime , divisibility , relatively prime

The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$,

2024 Abelkonkurransen Finale, 1a

Tags: modular arithmetic , coprime , coprime numbers , number theory

Determine all integers $n \ge 2$ such that $n \mid s_n-t_n$ where $s_n$ is the sum of all the integers in the interval $[1,n]$ that are mutually prime to $n$, and $t_n$ is the sum of the remaining integers in the same interval.

2016 Lusophon Mathematical Olympiad, 1

Tags: number theory , prime factor , product , coprime , positive integer

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

2012 Danube Mathematical Competition, 2

Tags: number theory , decimal representation , decimal , coprime

Consider the natural number prime $p, p> 5$. From the decimal number $\frac1p$, randomly remove $2012$ numbers, after the comma. Show that the remaining number can be represented as $\frac{a}{b}$ , where $a$ and $b$ are coprime numbers , and $b$ is multiple of $p$.

2003 All-Russian Olympiad Regional Round, 9.7

Tags: number theory , coprime , relatively prime

Prove that of any six four-digit numbers, mutual prime in total, you can always choose five numbers that are also relatively prime in total. [hide=original wording]Докажите, что из любых шести четырехзначных чисел, взаимно простых в совокупности, всегда можно выбратьпя ть чисел, также взаимно простых в совокупности.[/hide]

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , coprime , divide

Let $n$ be a positive integer and $A$ a set containing $8n + 1$ positive integers co-prime with $6$ and less than $30n$. Prove that there exist $a, b \in A$ two different numbers such that $a$ divides $b$.

2018 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , coprime , coprime numbers , sum of powers , divide

Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$

2007 Bulgarian Autumn Math Competition, Problem 10.3

Tags: number theory , coprime , sum function

For a natural number $m>1$ we'll denote with $f(m)$ the sum of all natural numbers less than $m$, which are also coprime to $m$. Find all natural numbers $n$, such that there exist natural numbers $k$ and $\ell$ which satisfy $f(n^{k})=n^{\ell}$.

2012 Greece JBMO TST, 2

Tags: number theory , least common multiple , coprime , diophantine equation

Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .

1998 Bundeswettbewerb Mathematik, 2

Tags: sequence , number theory , consecutive , coprime , perfect square

Prove that there exists an infinite sequence of perfect squares with the following properties: (i) The arithmetic mean of any two consecutive terms is a perfect square, (ii) Every two consecutive terms are coprime, (iii) The sequence is strictly increasing.

2004 All-Russian Olympiad Regional Round, 9.4

Tags: number theory , coprime

Three natural numbers are such that the product of any two of them is divided by the sum of these two numbers. Prove that these three numbers have a common divisor greater than one.

2022 Durer Math Competition (First Round), 4

Tags: number theory , coprime , combinatorics

We want to partition the integers $1, 2, 3, . . . , 100$ into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition? [i]We call two integers coprime if they have no common divisor greater than $1$.[/i]

1963 Polish MO Finals, 1

Tags: number theory , coprime , digtis

Prove that two natural numbers whose digits are all ones are relatively prime if and only if the numbers of their digits are relatively prime.

2006 Korea Junior Math Olympiad, 2

Tags: number theory , coprime , integer , positive integer

Find all positive integers that can be written in the following way $\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}$ . Also, $a,b, c$ are positive integers that are pairwise relatively prime.

2016 Bosnia and Herzegovina Team Selection Test, 4

Tags: number theory , additive number theory , coprime

Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: combinatorics , combinatorial number theory , coprime , set

Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that: every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number

2017 Tuymaada Olympiad, 6

Tags: number theory , sum of divisors , coprime

Let $\sigma(n)$ denote the sum of positive divisors of a number $n$. A positive integer $N=2^r b$ is given, where $r$ and $b$ are positive integers and $b$ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and $\sigma(b)$ are coprime. (J. Antalan, J. Dris)

2003 All-Russian Olympiad Regional Round, 10.7

Tags: coprime , number theory , combinatorics

Prove that from an arbitrary set of three-digit numbers, including at least four numbers that are mutually prime, you can choose four numbers that are also mutually prime