Found problems: 71
1965 Polish MO Finals, 2
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.
2008 Korea Junior Math Olympiad, 3
For all positive integers $n$, prove that there are integers $x, y$ relatively prime to $5$ such that $x^2 + y^2 = 5^n$.
2012 IMAR Test, 2
Given an integer $n \ge 2$, evaluate $\Sigma \frac{1}{pq}$ ,where the summation is over all coprime integers $p$ and $q$ such that $1 \le p < q \le n$ and $p + q > n$.
1979 Poland - Second Round, 5
Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.
1998 Bundeswettbewerb Mathematik, 2
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.
2007 Korea Junior Math Olympiad, 2
If $n$ is a positive integer and $a, b$ are relatively prime positive integers, calculate $(a + b,a^n + b^n)$.
2009 Estonia Team Selection Test, 2
Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer.
a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice.
b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?
2016 Saudi Arabia IMO TST, 1
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.
2015 IFYM, Sozopol, 6
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?
1997 Israel National Olympiad, 5
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.
1985 Polish MO Finals, 1
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 $n(n+17)$.
1969 IMO Longlists, 18
$(FRA 1)$ 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 $\omega$
2016 Lusophon Mathematical Olympiad, 1
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?
1999 Bundeswettbewerb Mathematik, 2
The sequences $(a_n)$ and $(b_n)$ are defined by $a_1 = b_1 = 1$ and $a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n$ for $n = 1,2,...$ Show that every two distinct terms of the sequence $(a_n)$ are coprime
2016 Balkan MO Shortlist, N1
Find all natural numbers $n$ for which $1^{\phi (n)} + 2^{\phi (n)} +... + n^{\phi (n)}$ is coprime with $n$.
2003 German National Olympiad, 6
Prove that there are infinitely many coprime, positive integers $a,b$ such that $a$ divides $b^2 -5$ and $b$ divides $a^2 -5.$
2009 Estonia Team Selection Test, 2
Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer.
a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice.
b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?
2017 Ecuador Juniors, 5
Two positive integers are coprime if their greatest common divisor is $1$. Let $C$ be the set of all divisors of the number $8775$ that are greater than $ 1$. A set of $k$ consecutive positive integers satisfies that each of them is coprime with some element of $C$. Determine the largest possible value of $K$.
2013 Danube Mathematical Competition, 2
Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.
2006 Korea Junior Math Olympiad, 2
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.
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.
1991 Romania Team Selection Test, 2
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.
1987 Brazil National Olympiad, 1
$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.
2011 Korea Junior Math Olympiad, 3
Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.
1989 Romania Team Selection Test, 2
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 $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$