Question 3
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a convex octagon such that no three diagonals are concurrent. The intersection of any two diagonals is called a “node.” Consider all convex quadrilaterals formed by four vertices of the octagon, called “subquadrilaterals.” Determine the smallest positive integer $n$ such that:
We can color $n$ “nodes” so that for all distinct $i,k\in\{1,2,3,4,5,6,7,8\}$, the numbers $s(i,k)$ are equal, where $s(i,k)$ denotes the number of “subquadrilaterals” having $A_i,A_k$ as vertices and whose diagonal intersection is a node.
We can color $n$ “nodes” so that for all distinct $i,k\in\{1,2,3,4,5,6,7,8\}$, the numbers $s(i,k)$ are equal, where $s(i,k)$ denotes the number of “subquadrilaterals” having $A_i,A_k$ as vertices and whose diagonal intersection is a node.