Problem

Source: 2018 China Southeast MO Grade 10 P2, Grade 11 P1

Tags: combinatorics



In a Cartesian plane, if both horizontal coordinate and vertical coordinate of a point are rational numbers, we call the point rational point. Otherwise, we call it irrational point. Consider an arbitrary regular pentagon on the Cartesian plane. Please compare the number of rational point and the number of irrational point among the five vertices of the pentagon. Prove your conclusion.