Problem

Source: Romania NMO - 2018

Tags: geometry, combinatorics



Let $n \in \mathbb{N}^*$ and consider a circle of length $6n$ along with $3n$ points on the circle which divide it into $3n$ arcs: $n$ of them have length $1,$ some other $n$ have length $2$ and the remaining $n$ have length $3.$ Prove that among these points there must be two such that the line that connects them passes through the center of the circle.