Found problems: 583
2016 Brazil Team Selection Test, 2
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2016$ good partitions.
PS. [url=https://artofproblemsolving.com/community/c6h1268855p6622233]2015 ISL C3 [/url] has 2015 instead of 2016
2019 CMIMC, 9
Let $a_0=29$, $b_0=1$ and $$a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1}$$ for $n\geq 1$. Determine the smallest positive integer $k$ for which $29$ divides $\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$.
2012 Hanoi Open Mathematics Competitions, 11
[b]Q11.[/b] Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$.
1993 China Team Selection Test, 1
Find all integer solutions to $2 x^4 + 1 = y^2.$
1913 Eotvos Mathematical Competition, 3
Let $d$ denote the greatest common divisor of the natural numbers $a$ and $b$, and let $d'$ denote the greatest common divisor of the natural numbers $a'$ and $b'$. Prove that $dd'$ is the greatest common divisor of the four numbers $$ aa' , \ \ ab' , \ \ ba' , \ \ bb' .$$
2016 Switzerland Team Selection Test, Problem 1
Let $n$ be a natural number. Two numbers are called "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?
2000 Romania Team Selection Test, 1
Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$.
[i]Mihai Pitticari & Sorin Rǎdulescu[/i]
2007 Singapore Junior Math Olympiad, 3
Let $n$ be a positive integer and $d$ be the greatest common divisor of $n^2+1$ and $(n + 1)^2 + 1$. Find all the possible values of $d$. Justify your answer.
1981 Brazil National Olympiad, 5
Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$.
2021 Peru IMO TST, P2
For any positive integers $a,b,c,n$, we define
$$D_n(a,b,c)=\mathrm{gcd}(a+b+c,a^2+b^2+c^2,a^n+b^n+c^n).$$
1. Prove that if $n$ is a positive integer not divisible by $3$, then for any positive integer $k$, there exist three integers $a,b,c$ such that $\mathrm{gcd}(a,b,c)=1$, and $D_n(a,b,c)>k$.
2. For any positive integer $n$ divisible by $3$, find all values of $D_n(a,b,c)$, where $a,b,c$ are three positive integers such that $\mathrm{gcd}(a,b,c)=1$.
2012 China Girls Math Olympiad, 3
Find all pairs $(a,b)$ of integers satisfying: there exists an integer $d \ge 2$ such that $a^n + b^n +1$ is divisible by $d$ for all positive integers $n$.
2007 Estonia National Olympiad, 4
Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third.
a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient.
b) Give an example of one of these larger numbers $a, b$ and $c$
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.
1990 AIME Problems, 10
The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?
2012 Kosovo Team Selection Test, 5
Prove that the equation
\[\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\]
has infinitly many natural solutions
2007 Bundeswettbewerb Mathematik, 1
Consider a regular polygon with 2007 vertices. The natural numbers $ 1,2, \ldots, 4014$ shall be distributed across the vertices and edge midpoints such that for each side the sum of its three numbers (two end points, one side center) has the same value. Show that such an assignment is possible.
2020 Latvia Baltic Way TST, 16
Given sequence $\{a_n\}$ satisfying:
$$ a_{n+1} = \frac{ lcm(a_n,a_{n-1})}{\gcd(a_n, a_{n-1})} $$
It is given that $a_{209} =209$ and $a_{361} = 361$. Find all possible values of $a_{2020}$.
1982 Dutch Mathematical Olympiad, 4
Determine $ \gcd (n^2\plus{}2,n^3\plus{}1)$ for $ n\equal{}9^{753}$.
2018 China National Olympiad, 1
Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying
$$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$
are both integers that are relatively prime to $p$. If $A_n$ is finite, let $f(n)$ denote $|A_n|$.
a) Prove that $A_n$ is finite if and only if $n \not = 2$.
b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that
$$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$
2009 Albania Team Selection Test, 3
Two people play a game as follows: At the beginning both of them have one point and in every move, one of them can double it's points, or when the other have more point than him, subtract to him his points. Can the two competitors have 2009 and 2002 points respectively? What about 2009 and 2003? Generally which couples of points can they have?
2021 JHMT HS, 3
Let $B=\{2^1,2^2,2^3,\dots,2^{21}\}.$ Find the remainder when
\[ \sum_{m, n \in B: \ m<n}\gcd(m,n) \]
is divided by $1000,$ where the sum is taken over all pairs of elements $(m,n)$ of $B$ such that $m<n.$
2013 Iran Team Selection Test, 8
Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$:
$a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$
1996 All-Russian Olympiad, 5
At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges?
[i]A. Shapovalov[/i]
2009 Harvard-MIT Mathematics Tournament, 4
Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.
2002 India IMO Training Camp, 9
On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on.
If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?