Olympiad problems

2008 Bundeswettbewerb Mathematik, 2

Let the positive integers $ a,b,c$ chosen such that the quotients $ \frac{bc}{b\plus{}c},$ $ \frac{ca}{c\plus{}a}$ and $ \frac{ab}{a\plus{}b}$ are integers. Prove that $ a,b,c$ have a common divisor greater than 1.

2022 Kyiv City MO Round 2, Problem 1

Tags: number theory , least common multiple

Find all triples $(a, b, c)$ of positive integers for which $a + [a, b] = b + [b, c] = c + [c, a]$. Here $[a, b]$ denotes the least common multiple of integers $a, b$. [i](Proposed by Mykhailo Shtandenko)[/i]

2013 Canadian Mathematical Olympiad Qualification Repechage, 3

Tags: least common multiple , number theory

A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.

PEN A Problems, 25

Tags: number theory , least common multiple , floor function , modular arithmetic , function , inequalities , logarithm

Show that ${2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)$ for all positive integers $n$.

2003 AMC 8, 19

Tags: number theory , least common multiple , modular arithmetic

How many integers between $1000$ and $2000$ have all three of the numbers $15$, $20$, and $25$ as factors? $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2018-IMOC, N2

Tags: number theory , least common multiple , greatest common divisor , fe , functional equation

Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$ for all $x,y\in\mathbb N$.

2020 Durer Math Competition Finals, 3

Tags: number theory , least common multiple , perfect square , consecutive

Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?

2005 Romania National Olympiad, 3

Tags: linear algebra , matrix , calculus , function , number theory , greatest common divisor , least common multiple

Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.