Olympiad problems

Mexican Quarantine Mathematical Olympiad, #2

Tags: relation , combinatorics

Let $n$ be an integer greater than $1$. A certain school has $1+2+\dots+n$ students and $n$ classrooms, with capacities for $1, 2, \dots, n$ people, respectively. The kids play a game in $k$ rounds as follows: in each round, when the bell rings, the students distribute themselves among the classrooms in such a way that they don't exceed the room capacities, and if two students shared a classroom in a previous round, they cannot do it anymore in the current round. For each $n$, determine the greatest possible value of $k$. [i]Proposed by Victor Domínguez[/i]

2014 IFYM, Sozopol, 6

Tags: relation , set , combinatorics

Let $A$ and $B$ be two non-infinite sets of natural numbers, each of which contains at least 3 elements. Two numbers $a\in A$ and $b\in B$ are called [i]"harmonious"[/i], if they are not coprime. It is known that each element from $A$ is not [i]harmonious[/i] with at least one element from $B$ and each element from $B$ is harmonious with at least one from $A$. Prove that there exist $a_1,a_2\in A$ and $b_1,b_2\in B$ such that $(a_1,b_1)$ and $(a_2,b_2)$ are [i]harmonious[/i] but $(a_1,b_2)$ and $(a_2,b_1)$ are not.

2019 India PRMO, 14

Tags: prime , perfect square , relation

Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.