Found problems: 583
1974 IMO Shortlist, 7
Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$
2010 China Northern MO, 7
Find all positive integers $x, y, z$ that satisfy the conditions: $$[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1$$
The symbols $[m,n]$ and $(m,n)$ respectively represent positive integers, the least common multiple and the greatest common divisor of $m$ and $n$.
2004 South africa National Olympiad, 1
Let $a=1111\dots1111$ and $b=1111\dots1111$ where $a$ has forty ones and $b$ has twelve ones. Determine the greatest common divisor of $a$ and $b$.
1990 USAMO, 3
Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \[ \{ n, n+1, n+2, \dots, n+32 \} \] so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)
1998 South africa National Olympiad, 5
Prove that \[ \gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)} \] is a prime if $n$ is a power of a prime, and 1 otherwise.
2013 Kazakhstan National Olympiad, 2
Prove that for all natural $n$ there exists $a,b,c$ such that $n=\gcd (a,b)(c^2-ab)+\gcd (b,c)(a^2-bc)+\gcd (c,a)(b^2-ca)$.
2010 Balkan MO, 4
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
2021 Simon Marais Mathematical Competition, A2
Define the sequence of integers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$, and
\[ a_{n+1} = \left(n+1-\gcd(a_n,n) \right) \times a_n \]
for all integers $n \ge 1$.
Prove that $\frac{a_{n+1}}{a_n}=n$ if and only if $n$ is prime or $n=1$.
[i]Here $\gcd(s,t)$ denotes the greatest common divisor of $s$ and $t$.[/i]
2020 Mediterranean Mathematics Olympiad, 1
Determine all integers $m\ge2$ for which there exists an integer $n\ge1$ with
$\gcd(m,n)=d$ and $\gcd(m,4n+1)=1$.
[i]Proposed by Gerhard Woeginger, Austria[/i]
2023 Austrian MO Regional Competition, 4
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
PEN O Problems, 39
Find the smallest positive integer $n$ for which there exist $n$ different positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying [list] [*] $\text{lcm}(a_1,a_2,\cdots,a_n)=1985$,[*] for each $i, j \in \{1, 2, \cdots, n \}$, $gcd(a_i,a_j)\not=1$, [*] the product $a_{1}a_{2} \cdots a_{n}$ is a perfect square and is divisible by $243$, [/list] and find all such $n$-tuples $(a_{1}, \cdots, a_{n})$.
2014 Czech-Polish-Slovak Junior Match, 1
The set of $\{1,2,3,...,63\}$ was divided into three non-empty disjoint sets $A,B$. Let $a,b,c$ be the product of all numbers in each set $A,B,C$ respectively and finally we have determined the greatest common divisor of these three products. What was the biggest result we could get?
2009 Princeton University Math Competition, 1
If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.
2016 Israel Team Selection Test, 4
Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.
2014 Contests, 3
Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$
2016 Canadian Mathematical Olympiad Qualification, 6
Determine all ordered triples of positive integers $(x, y, z)$ such that $\gcd(x+y, y+z, z+x) > \gcd(x, y, z)$.
2024 Macedonian TST, Problem 6
Let \(a,b\) be positive integers such that \(a+1\), \(b+1\), and \(ab\) are perfect squares. Prove that $\gcd(a,b)+1$ is also a perfect square.
2016 Iran Team Selection Test, 6
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/i]
2013 AMC 8, 10
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660$
2015 Mathematical Talent Reward Programme, SAQ: P 5
Let $a$ be the smallest and $A$ the largest of $n$ distinct positive integers. Prove that the least common multiple of these numbers is greater than or equal to $n a$ and that the greatest common divisor is less than or equal to $\frac{A}{n}$
2006 China Team Selection Test, 3
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$.
2010 Indonesia TST, 4
Prove that for all integers $ m$ and $ n$, the inequality
\[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\]
holds.
[i]Nanang Susyanto, Jogjakarta [/i]
2003 India IMO Training Camp, 2
Find all triples $(a,b,c)$ of positive integers such that
(i) $a \leq b \leq c$;
(ii) $\text{gcd}(a,b,c)=1$; and
(iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.
2019 BMT Spring, 17
Let $C$ be a circle of radius $1$ and $O$ its center. Let $\overline{AB}$ be a chord of the circle and $D$ a point on $\overline{AB}$ such that $OD =\frac{\sqrt2}{2}$ such that $D$ is closer to $ A$ than it is to $ B$, and if the perpendicular line at $D$ with respect to $\overline{AB}$ intersects the circle at $E $and $F$, $AD = DE$. The area of the region of the circle enclosed by $\overline{AD}$, $\overline{DE}$, and the minor arc $AE$ may be expressed as $\frac{a + b\sqrt{c} + d\pi}{e}$ where $a, b, c, d, e$ are integers, gcd $(a, b, d, e) = 1$, and $c$ is squarefree. Find $a + b + c + d + e$
2014 JHMMC 7 Contest, 10
Find the sum of the greatest common factor and the least common multiple of $12$ and $18$.