S + V = H + 3 in a closed rectangular neighborhood
Source: Flanders Math Olympiad 2017 p3
December 24, 2022
combinatorics
Problem Statement
In a closed rectangular neighborhood there are:
streets (these are straight roads of maximum length),
four-arm intersections ( https://cdn.artofproblemsolving.com/attachments/e/4/6a5974a30dc182b59a519a8ef4eb4f1412e05e.png),
city blocks (these are rectangular areas bounded by four streets, which are no be intersected by another street) and
represents the number of -intersections (https://cdn.artofproblemsolving.com/attachments/0/a/b390a30a0b27d83db681f70f633bdeed697163.png ).
For example, in the neighborhood below, there are streets, four-arm intersections, city blocks and -intersections.
https://cdn.artofproblemsolving.com/attachments/a/2/c1a5e463d0fb5671ac0702c91cfc2272d4e2c3.png
Prove that in each district .