Question 33
In each vertex of a regular $50$-gon one of the numbers $1$ or $2$ is written. No three consecutive vertices have the same number written in them. The number $1$ is written twenty times and the number $2$ thirty times in total. For each vertex the product of the numbers written in two adjacent vertices and the number in that vertex is calculated, and all such products are added up.
Prove that the result is always the same and determine it.
Prove that the result is always the same and determine it.