Choose $3n$ points on a circle, show that there are two diametrically opposite point > On a circle of length $6n$, we choose $3n$ points such that they split the circle into $n$ arcs of length $1$, $n$ arcs of length $2$, $n$ arcs of length $3$. Show that there exists two chosen points which are diametrically opposite. Source: * Russian MO $1982$ * Swiss MO $2006$ \- Final round * IMAC $2012$ * Romania MO $2018$ \- $9$. grade * * * Edit: Partition of circumference into $3k$ arcs

Well, that's simple. If we want to avoid having opposite points, then obviously every 1-arc must be positioned diametrically opposite to the middle of some 3-arc. Consider one such pair. What is the length of the great arc connecting their edges? Apparently, $3n-2$, which means it has the **same** parity as $n$. But it is composed of $n-1$ odd arcs (1s or 3s) and some unknown number of 2-arcs which do not matter, so it must have the **opposite** parity.

That's it.