Consider the following problem posed by Michael Khoury, U.S. Math Olympiad Team Member, in “The Problem Solving Competition,” Oklahoma Publishing Company and the American Society for the Communication...

Consider the following problem posed by Michael Khoury, U.S. Math Olympiad Team Member, in “The Problem Solving Competition,” Oklahoma Publishing Company and the American Society for the Communication of Mathematics, February 1999. Bob is teaching a class with n students. There are n desks in the classroom numbered from 1 to n. Bob has prepared a seating chart, but the students have already seated themselves randomly. Bob calls off the name of the person who belongs in seat 1. This person vacates the seat he or she is currently occupying and takes his or her rightful seat. If this displaces a person already in the seat, that person stands at the front of the room until he or she is assigned a seat. Bob does this for each seat in turn. Let X be the number of students standing at the front of the room after k, 1 ≤ k