This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 71

2014 Danube Mathematical Competition, 3
Tags: arithmetic sequence , arithmetic progression , coprime , number theory
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.

2024 Abelkonkurransen Finale, 1a
Tags: coprime numbers , coprime , number theory , modular arithmetic
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.

1979 Poland - Second Round, 5
Tags: coprime , construction , number theory
Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.

2016 Lusophon Mathematical Olympiad, 1
Tags: positive integer , prime factor , product , number theory , coprime
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?

2024 Dutch IMO TST, 1
Tags: divisor , coprime , number theory
For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

2016 Saudi Arabia IMO TST, 1
Tags: sum , coprime , number theory
Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.

1997 Israel National Olympiad, 5
Tags: prime , number theory , coprime
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.

2015 Saudi Arabia BMO TST, 4
Tags: prime , coprime , gcd , number theory
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

2018 Rioplatense Mathematical Olympiad, Level 3, 3
Tags: coprime numbers , sum of powers , divide , number theory , coprime
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}$

2018 Dutch BxMO TST, 2
Tags: integer , sides of a triangle , coprime , geometry
Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

2012 Greece JBMO TST, 2
Tags: coprime , diophantine equation , least common multiple , number theory
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 $p^2,q^2$ .

1965 Polish MO Finals, 2
Tags: coprime , integer , number theory , algebra
Prove that if the numbers $ x_1 $ and $ x_2 $ are roots of the equation $ x^2 + px - 1 = 0 $, where $ p $ is an odd number, then for every natural $n$number $ x_1^n + x_2^n $ and $ x_1^{n+1} + x_2^{n+1} $ are integer and coprime.

1969 IMO Longlists, 18
Tags: coprime , frobenius , additive number theory , 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 $\omega$

2014 IFYM, Sozopol, 5
Tags: polynomial , coprime , number theory
Let $f(x)$ be a polynomial with integer coefficients, for which there exist $a,b\in \mathbb{Z}$ ($a\neq b$), such that $f(a)$ and $f(b)$ are coprime. Prove that there exist infinitely many values for $x$, such that each $f(x)$ is coprime with any other.

2010 Estonia Team Selection Test, 1
Tags: prime , gcd , power of prime , coprime , number theory
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

1985 Polish MO Finals, 1
Tags: coprime , maximum , number theory
Find the largest $k$ such that for every positive integer $n$ we can find at least $k$ numbers in the set $\{n+1, n+2, ... , n+16\}$ which are coprime with $n(n+17)$.

2015 IFYM, Sozopol, 6
Tags: number theory , coprime
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?

2023 Indonesia TST, N
Tags: gcd , integer sequence , infinite sequence , coprime , number theory
Given an integer $a>1$. Prove that there exists a sequence of positive integers \[ n_1, n_2, n_3, \ldots \] Such that \[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.

2019 Junior Balkan Team Selection Tests - Romania, 2
Tags: coprime , divide , number theory
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 $b$.

2004 All-Russian Olympiad Regional Round, 9.4
Tags: coprime , number theory
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.

1991 Romania Team Selection Test, 2
Tags: recurrence relation , sequence , coprime , number theory
The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer. Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.

1989 Romania Team Selection Test, 2
Tags: coprime , number theory , system of equations , algebra
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 $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

2006 Korea Junior Math Olympiad, 2
Tags: positive integer , integer , coprime , number theory
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.

2015 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: coprime , number theory
For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$

1988 Mexico National Olympiad, 5
Tags: number theory , coprime , greatest common divisor
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.