Olympiad problems

2012 Canada National Olympiad, 2

For any positive integers $n$ and $k$, let $L(n,k)$ be the least common multiple of the $k$ consecutive integers $n,n+1,\ldots ,n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n,k)>bL(n+1,k)$.

2010 NZMOC Camp Selection Problems, 4

Tags: least common multiple , number theory , greatest common divisor , lcm , gcd

Find all positive integer solutions $(a, b)$ to the equation $$\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)}$$ for (i) $n = 2007$; (ii) $n = 2010$.

1964 Putnam, B5

Tags: least common multiple , convergence

Let $u_n$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers. Prove that the series $$\sum_{n=1}^{\infty} \frac{1}{ u_n }$$ is convergent

1978 AMC 12/AHSME, 27

Tags: number theory , least common multiple

There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 11$, has a remainder of $1$. What is the difference between the two smallest such integers? $\textbf{(A) }2310\qquad\textbf{(B) }2311\qquad\textbf{(C) }27,720\qquad\textbf{(D) }27,721\qquad \textbf{(E) }\text{none of these}$

OMMC POTM, 2022 9

Tags: least common multiple , inequalities , number theory

For positive integers $a_1 < a_2 < \dots < a_n$ prove that $$\frac{1}{\operatorname{lcm}(a_1, a_2)}+\frac{1}{\operatorname{lcm}(a_2, a_3)}+\dots+\frac{1}{\operatorname{lcm}(a_{n-1}, a_n)} \leq 1-\frac{1}{2^{n-1}}.$$ [i]Proposed by Evan Chang (squareman), USA[/i]

2011 NIMO Summer Contest, 11

Tags: number theory , least common multiple , greatest common divisor

How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

2014 NZMOC Camp Selection Problems, 6

Tags: number theory , least common multiple , lcm

Determine all triples of positive integers $a$, $ b$ and $c$ such that their least common multiple is equal to their sum.

2007 Junior Balkan Team Selection Tests - Romania, 4

Tags: least common multiple , number theory

Find all integer positive numbers $n$ such that: $n=[a,b]+[b,c]+[c,a]$, where $a,b,c$ are integer positive numbers and $[p,q]$ represents the least common multiple of numbers $p,q$.