Question 74
Let $n \ge 2$ 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 pairs of superimposed sectors have different colors. Prove that these Welsh darts boards can be matched so that at least $2\left\lfloor\frac{n}{2}\right\rfloor + 2$ pairs of superimposed sectors have the same color.