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.
1982 Kurschak Competition, 2
Prove that for any integer $k > 2$, there exist infinitely many positive integers $n$ such that the least common multiple of $n$, $n + 1$,$...$, $n + k - 1$ is greater than the least common multiple of $n + 1$,$n + 2$,$...$, $n + k$.
2018 Switzerland - Final Round, 3
Determine all natural integers $n$ for which there is no triplet $(a, b, c)$ of natural numbers such that:
$$n = \frac{a \cdot \,\,lcm(b, c) + b \cdot lcm \,\,(c, a) + c \cdot lcm \,\, (a, b)}{lcm \,\,(a, b, c)}$$
2008 Thailand Mathematical Olympiad, 7
Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.
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 $n$.
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 $2011$.
2008 Postal Coaching, 1
Prove that for any $n \ge 1$,
$LCM _{0\le k\le n} \big \{$ $n \choose k$ $\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}$
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?
2020 Tournament Of Towns, 2
Alice had picked positive integers $a, b, c$ and then tried to find positive integers $x, y, z$ such that $a = lcm (x, y)$, $b = lcm(x, z)$, $c = lcm(y, z)$. It so happened that such $x, y, z$ existed and were unique. Alice told this fact to Bob and also told him the numbers $a$ and $b$. Prove that Bob can find $c$. (Note: lcm = least common multiple.)
Boris Frenkin
2016 Switzerland - Final Round, 9
Let $n \ge 2$ be a natural number. For an $n$-element subset $F$ of $\{1, . . . , 2n\}$ we define $m(F)$ as the minimum of all $lcm \,\, (x, y)$ , where $x$ and $y$ are two distinct elements of $F$. Find the maximum value of $m(F)$.
2024 Czech and Slovak Olympiad III A, 1
Let $a, b, c$ be positive integers such that one of the values $$gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b)$$
is equal to the product of the remaining two. Prove that one of the numbers $a, b, c$ is a multiple of another of them.
2010 Brazil Team Selection Test, 2
Let $k > 1$ be a fixed integer. Prove that there are infinite positive integers $n$ such that
$$ lcm \, (n, n + 1, n + 2, ... , n + k) > lcm \, (n + 1, n + 2, n + 3,... , n + k + 1).$$
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.
2016 India PRMO, 15
Find the number of pairs of positive integers $(m; n)$, with $m \le n$, such that the ‘least common multiple’ (LCM) of $m$ and $n$ equals $600$.
2019 Caucasus Mathematical Olympiad, 2
Determine if there exist five consecutive positive integers such that their LCM is a perfect square.
2002 Estonia National Olympiad, 1
The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.
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?
2010 Czech And Slovak Olympiad III A, 6
Find the minimum of the expression $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.
2017 Purple Comet Problems, 4
Find the least positive integer $m$ such that $lcm(15,m) = lcm(42,m)$. Here $lcm(a, b)$ is the least common multiple of $a$ and $b$.
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 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 $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$
2021 Lotfi Zadeh Olympiad, 3
Find the least possible value for the fraction
$$\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)}$$
over all distinct positive integers $a, b, c$.
By $lcm(x, y)$ we mean the least common multiple of $x, y$ and by $gcd(x, y)$ we mean the greatest common divisor of $x, y$.
2005 Cuba MO, 7
Determine all triples of positive integers $(x, y, z)$ that satisfy
$$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \
lcm(x, y, z) = 2400.$$
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 $k$.
Kvant 2019, M2566
Determine if there exist five consecutive positive integers such that their LCM is a perfect square.