2009 ToT Spring Senior O P4 zeros and ones written in a row
Source:
March 7, 2020
combinatorics
Problem Statement
Several zeros and ones are written down in a row. Consider all pairs of digits (not necessarily adjacent) such that the left digit is while the right digit is . Let be the number of the pairs in which and are separated by an even number of digits (possibly zero), and let be the number of the pairs in which and are separated by an odd number of digits. Prove that .