Found problems: 51
2004 Switzerland - Final Round, 8
A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers $a, b$, wipe them out and instead write gcd$(a, b)$ and lcm$(a, b)$. Show that the content of the list no longer changed after a certain point in time.
2020 MMATHS, I5
For some positive integers $m>n$, the quantities $a=\text{lcm}(m,n)$ and $b=\gcd(m,n)$ satisfy $a=30b$. If $m-n$ divides $a$, then what is the value of $\frac{m+n}{b}$?
[i]Proposed by Andrew Wu[/i]
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 $n$.
2019 Durer Math Competition Finals, 11
What is the smallest possible value of the least common multiple of $a, b, c, d$ if we know that these four numbers are distinct and $a + b + c + d = 1000$?
2015 Junior Balkan Team Selection Tests - Moldova, 8
Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities:
$$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$
([x, y]represents the least common multiple of positive integers x,y)
2021 Durer Math Competition (First Round), 4
Determine all triples of positive integers $a, b, c$ that satisfy
a) $[a, b] + [a, c] + [b, c] = [a, b, c]$.
b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$.
Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.
2021 Czech-Polish-Slovak Junior Match, 3
Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.
2013 Tournament of Towns, 3
Denote by $[a, b]$ the least common multiple 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 Kyiv City MO Round 2, Problem 1
a) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+2d]$?
b) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+4d]$?
Here $[a, b]$ denotes the least common multiple of integers $a, b$.
2023 Azerbaijan JBMO TST, 1
Let $a < b < c < d < e$ be positive integers. Prove that
$$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$
where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?
2004 Estonia Team Selection Test, 5
Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.
2014 NZMOC Camp Selection Problems, 6
Determine all triples of positive integers $a$, $ b$ and $c$ such that their least common multiple is equal to their sum.
2022 JBMO Shortlist, N2
Let $a < b < c < d < e$ be positive integers. Prove that
$$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$
where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?
2008 Postal Coaching, 1
Prove that for any $n \ge 1$,
$LCM _{0\le k\le n} \big \{$ $n \choose k$ $\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}$
2019 Caucasus Mathematical Olympiad, 2
Determine if there exist five consecutive positive integers such that their LCM is a perfect square.
2001 Singapore Team Selection Test, 3
Let $L(n)$ denote the least common multiple of $\{1, 2 . . . , n\}$.
(i) Prove that there exists a positive integer $k$ such that $L(k) = L(k + 1) = ... = L(k + 2000)$.
(ii) Find all $m$ such that $L(m + i) \ne L(m + i + 1)$ for all $i = 0, 1, 2$.
2013 Saudi Arabia BMO TST, 2
For positive integers $a$ and $b$, $gcd (a, b)$ denote their greatest common divisor and $lcm (a, b)$ their least common multiple. Determine the number of ordered pairs (a,b) of positive integers satisfying the equation $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$
2004 Estonia National Olympiad, 1
Find all triples of positive integers $(x, y, z)$ satisfying $x < y < z$, $gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8$ and $lcm(x, y,z) = 2400$.
2013 Switzerland - Final Round, 1
Find all triples $(a, b, c)$ of natural numbers such that the sets
$$\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\}$$ and
$$\{2, 3, 5, 30, 60\}$$
are the same.
Remark: For example, the sets $\{1, 2013\}$ and $\{1, 1, 2013\}$ are equal.
2010 Brazil Team Selection Test, 2
Let $k > 1$ be a fixed integer. Prove that there are infinite positive integers $n$ such that
$$ lcm \, (n, n + 1, n + 2, ... , n + k) > lcm \, (n + 1, n + 2, n + 3,... , n + k + 1).$$
Kvant 2019, M2566
Determine if there exist five consecutive positive integers such that their LCM is a perfect square.
1998 Tournament Of Towns, 1
(a) Prove that for any two positive integers a and b the equation $lcm (a, a + 5) = lcm (b, b + 5)$ implies $a = b$.
(b) Is it possible that $lcm (a, b) = lcm (a + c, b + c)$ for positive integers $a, b$ and $c$?
(A Shapovalov)
PS. part (a) for Juniors, both part for Seniors
2005 Cuba MO, 7
Determine all triples of positive integers $(x, y, z)$ that satisfy
$$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \
lcm(x, y, z) = 2400.$$
2011 Cuba MO, 7
Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.
1999 Swedish Mathematical Competition, 6
$S$ is any sequence of at least $3$ positive integers. A move is to take any $a, b$ in the sequence such that neither divides the other and replace them by gcd $(a,b)$ and lcm $(a,b)$. Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.