Found problems: 71
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 $n(n+17)$.
2024 Dutch IMO TST, 1
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 $\alpha(n)=\beta(n)$.
2009 Bulgaria National Olympiad, 1
The natural numbers $a$ and $b$ satis fy the inequalities $a > b > 1$ . It is also known that the equation
$\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$.
Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).
2010 Estonia Team Selection Test, 1
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.
2017 Irish Math Olympiad, 1
Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?
1993 Abels Math Contest (Norwegian MO), 3
The Fermat-numbers are defined by $F_n = 2^{2^n}+1$ for $n\in N$.
(a) Prove that $F_n = F_{n-1}F_{n-2}....F_1F_0 +2$ for $n > 0$.
(b) Prove that any two different Fermat numbers are coprime
2015 Costa Rica - Final Round, N4
Show that there are no triples $(a, b, c)$ of positive integers such that
a) $a + c, b + c, a + b$ do not have common multiples in pairs.
b)$\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}$ are integer numbers.
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 $a^2 -5.$
2011 Korea Junior Math Olympiad, 6
For a positive integer $n$, define the set $S_n$ as $S_n =\{(a, b)|a, b \in N, lcm[a, b] = n\}$ . Let $f(n)$ be the sum of $\phi (a)\phi (b)$ for all $(a, b) \in S_n$. If a prime $p$ relatively prime to $n$ is a divisor of $f(n)$, prove that there exists a prime $q|n$ such that $p|q^2 - 1$.
2011 Regional Olympiad of Mexico Center Zone, 3
We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $.
2011 IMAR Test, 4
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
2010 Estonia Team Selection Test, 1
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.
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 $n$?
2024 Brazil Team Selection Test, 1
Given an integer $n > 1$, let $1 = a_1 < a_2 < \cdots < a_t = n - 1$ be all positive integers less than $n$ that are coprime to $n$. Find all $n$ such that there is no $i \in \{1, 2, \ldots , t - 1\}$ satisfying $3 | a_i + a_{i+1}$.
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.
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 $n$?
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.
2014 Danube Mathematical Competition, 3
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.
2007 Korea Junior Math Olympiad, 2
If $n$ is a positive integer and $a, b$ are relatively prime positive integers, calculate $(a + b,a^n + b^n)$.
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 $k(p^s) = p^{n(s-1)} k(p)$.
2008 Korea Junior Math Olympiad, 3
For all positive integers $n$, prove that there are integers $x, y$ relatively prime to $5$ such that $x^2 + y^2 = 5^n$.
2015 Bosnia And Herzegovina - Regional Olympiad, 2
For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$
2012 Singapore Junior Math Olympiad, 5
Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number.
(Note: Two positive integers $m, n$ are coprime if their only common factor is 1)
2023 Indonesia TST, N
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 $i \neq j$.
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 $\omega$