Question 77
Let $n\geq3$ be an integer and mark $n+1$ equally spaced points on a circle. Consider all labelings with $0,1,\ldots,n$, each used once; labelings coinciding after a rotation are identical. A labeling is beautiful if for all $a\lt b\lt c\lt d$ with $a+d=b+c$, the chord joining $a,d$ does not intersect the chord joining $b,c$. Let $M$ be the number of beautiful labelings and $N$ the number of ordered pairs $(x,y)$ of positive integers with $x+y\leq n$ and $\gcd(x,y)=1$. Prove $M=N+1$.