Question 54
Let $n\ge3$ be an integer. Mark $n+1$ equally spaced points on a circle and label them with $0,1,\dots,n$, using each number exactly once. Two labellings are regarded as identical if a rotation takes one to the other. A labelling is beautiful if for any four labels $a\lt b\lt c\lt d$ satisfying $a+d=b+c$, the chord joining $a,d$ does not cross the chord joining $b,c$. Let $M$ count beautiful labellings, and let $N$ count ordered pairs $(x,y)$ of positive integers satisfying $x+y\le n$ and $\gcd(x,y)=1$. Prove $M=N+1$.