Found problems: 583
2011 Czech-Polish-Slovak Match, 2
Written on a blackboard are $n$ nonnegative integers whose greatest common divisor is $1$. A [i]move[/i] consists of erasing two numbers $x$ and $y$, where $x\ge y$, on the blackboard and replacing them with the numbers $x-y$ and $2y$. Determine for which original $n$-tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where $n-1$ of the numbers of the blackboard are zeroes.
2009 China Team Selection Test, 3
Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$
2016 AMC 12/AHSME, 24
There are exactly $77,000$ ordered quadruples $(a,b,c,d)$ such that $\gcd(a,b,c,d)=77$ and $\operatorname{lcm}(a,b,c,d)=n$. What is the smallest possible value of $n$?
$\textbf{(A)}\ 13,860 \qquad
\textbf{(B)}\ 20,790 \qquad
\textbf{(C)}\ 21,560 \qquad
\textbf{(D)}\ 27,720 \qquad
\textbf{(E)}\ 41,580$
2011 Middle European Mathematical Olympiad, 8
We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality
\[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\]
holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i].
[b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.
1953 Moscow Mathematical Olympiad, 244
Prove that $gcd (a + b, lcm(a, b)) = gcd (a, b)$ for any $a, b$.
2012 Silk Road, 3
Let $n > 1$ be an integer.
Determine the greatest common divisor of the set of numbers $\left\{ \left( \begin{matrix}
2n \\
2i+1 \\
\end{matrix} \right):0 \le i \le n-1 \right\}$
i.e. the largest positive integer, dividing $\left( \begin{matrix}
2n \\
2i+1 \\
\end{matrix} \right)$ without remainder for every $i = 0, 1, ..., n–1$ .
(Here $\left( \begin{matrix}
m \\
l \\
\end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}$ is binomial coefficient.)
2018 IFYM, Sozopol, 2
$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If
$d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $
find all possible values of $d$.
2009 USAMO, 6
Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$ Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.
2022 Baltic Way, 18
Find all pairs $(a, b)$ of positive integers such that $a \le b$ and
$$ \gcd(x, a) \gcd(x, b) = \gcd(x, 20) \gcd(x, 22) $$
holds for every positive integer $x$.
2001 Saint Petersburg Mathematical Olympiad, 11.4
For any two positive integers $n>m$ prove the following inequality:
$$[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}$$
As always, $[x,y]$ means the least common multiply of $x,y$.
[I]Proposed by A. Golovanov[/i]
2011 Kyiv Mathematical Festival, 1
Solve the equation $m^{gcd(m,n)} = n^{lcm(m,n)}$ in positive integers, where gcd($m, n$) – greatest common
divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.
1988 IMO Longlists, 65
The Fibonacci sequence is defined by \[ a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1. \] Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.
1996 Rioplatense Mathematical Olympiad, Level 3, 6
Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies:
(i) $f(1995) = 1996$
(ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$.
Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ .
2009 Saint Petersburg Mathematical Olympiad, 1
$x,y$ are naturals. $GCM(x^7,y^4)*GCM(x^8,y^5)=xy$ Prove that $xy$ is cube
2005 Thailand Mathematical Olympiad, 9
Compute gcd $\left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right)$
2007 Harvard-MIT Mathematics Tournament, 22
The sequence $\{a_n\}_{n\geq 1}$ is defined by $a_{n+2}=7a_{n+1}-a_n$ for positive integers $n$ with initial values $a_1=1$ and $a_2=8$. Another sequence, $\{b_n\}$, is defined by the rule $b_{n+2}=3b_{n+1}-b_n$ for positive integers $n$ together with the values $b_1=1$ and $b_2=2$. Find $\gcd(a_{5000},b_{501})$.
1996 Dutch Mathematical Olympiad, 5
For the positive integers $x , y$ and $z$ apply $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ .
Prove that if the three numbers $x , y,$ and $z$ have no common divisor greater than $1$, $x + y$ is the square of an integer.
2024 Mozambique National Olympiad, P1
Among families in a neighborhood in the city of Chimoio, a total of $144$ notebooks, $192$ pencils and $216$ erasers were distributed. This distribution was made so that the largest possible number of families was covered and everyone received the same number of each material, without having any leftovers. In this case, how many notebooks, pencils and erasers did each family receive?
2013 Tournament of Towns, 3
Denote by $(a, b)$ the greatest common divisor of $a$ and $b$.
Let $n$ be a positive integer such that $(n, n + 1) < (n, n + 2) <... < (n,n + 35)$. Prove that $(n, n + 35) < (n,n + 36)$.
2022 Bolivia IMO TST, P4
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2014 Baltic Way, 19
Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\]
2012 AMC 8, 13
Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$1.43$. Sharona bought some of the same pencils and paid $\$1.87$. How many more pencils did Sharona buy than Jamar?
$\textbf{(A)}\hspace{.05in}2 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}4 \qquad \textbf{(D)}\hspace{.05in}5 \qquad \textbf{(E)}\hspace{.05in}6 $
2023 Bulgaria EGMO TST, 2
Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.
2008 IMO Shortlist, 3
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
2015 IMO Shortlist, C3
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 $2015$ good partitions.