Let $n\geqslant 2$ be an integer. A Welsh darts board is a disc divided into $2n$ equal sectors, half of them being red and the other half being white. Two Welsh darts boards are matched if they have the same radius and they are superimposed so that each sector of the first board comes exactly over a sector of the second board. Suppose that two given Welsh darts boards can be matched so that more than half of the paurs of superimposed sectors have different colours. Prove that these Welsh darts boards can be matched so that at least $2\lfloor n/2\rfloor +2$ pairs of superimposed sectors have the same colour.
Problem
Source: Romania JBMO TST 2024 Day 2 P4
Tags: combinatorics
Sadigly
12.01.2025 21:19
Bump....
Double07
13.01.2025 00:21
This is just a generalization of Canada 2009/2. Consider all $2n$ rotations of one of the darts board. We will compute the expected value of $A$, the number of pairs of superimposed sectors have the same colour. Each sector will be paired in exactly $n$ of the $2n$ rotations, so each sector has probability $\dfrac{1}{2}$ to be paired with a same-coloured sector. This implies $\mathbb{E}[A]=\dfrac{2n}{2}=n$. But we also know that there exists a rotation for which $A<n$, so there exists a rotation for which $A>n$. We can also easily prove that $A$ is even
Let there be $r$ pairs of superimposed red sectors and $w$ pairs of superimposed white sectors, $w+r=A$.
So there are $k=2n-A$ pairs of superimposed diffrent-coloured sectors, in which there are exactly one white and one red sector.
This means there are $2n=k+2\cdot r$ red sectors in total, so $k$ is even, which implies $A$ is even.
so $A$ must be at least the lowest even number larger than $n$, which is $2\lfloor n/2\rfloor+2$, and we're done.