AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_21 / If A and B are vertices of a polyhedron, define the distance d(A,B) to be the minimum number of edges of the polyhedron one must traverse in order to connect A and B. For example, …

If A and B are vertices of a polyhedron, define the distance d(A,B) to be the minimum number of edges of the polyhedron one must traverse in order to connect A and B. For example, if (AB)¯ is an edge of the polyhedron, then d(A,B)=1, but if (AC)¯ and (CB)¯ are edges and (AB)¯ is not an edge, then d(A,B)=2. Let Q, R, and S be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). Find the probability that d(Q,R)>d(R,S). The final answer can be written in the form (m)/(n), where m and n are relatively prime positive integers. What is m+n?

If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). Find the probability that $d(Q, R) > d(R, S)$. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?