Four race car drivers participate in a race on a loop track. All four are going at a constant speed. Assume that they did a flying start. That is, all four crossed the start line at the same instant while each was going their constant speed. Then they continue driving forever and it is the case that for any three of the cars there is a moment in time, after the start, when these three cars are located at the same point along the track (all three are passing each other). Prove that there is a moment in time, after the start, when all four cars are located at the same point along the track.
To prove that there is a moment in time when all four cars are located at the same point along the track, we can use the Pigeonhole Principle.
The Pigeonhole Principle states that if you distribute more objects into fewer containers, some container must contain more than one object.
Let's consider the possible scenarios when three cars are passing each other at a given moment in time.
1. Car A, Car B, Car C: At a certain time, Car A is passing Car B, Car B is passing Car C, and Car C is passing Car A.
2. Car A, Car B, Car D: At a certain time, Car A is passing Car B, Car B is passing Car D, and Car D is passing Car A.
3. Car A, Car C, Car D: At a certain time, Car A is passing Car C, Car C is passing Car D, and Car D is passing Car A.
4. Car B, Car C, Car D: At a certain time, Car B is passing Car C, Car C is passing Car D, and Car D is passing Car B.
Now, let's analyze the scenarios.
In each case, three cars are passing each other at the same moment, which means they are all located at the same point along the track.
We can treat each of these scenarios as a "container" and the cars as "objects" being distributed into these containers. Since we have four cars, and each scenario involves three cars, there must be at least two scenarios that share the same set of cars.
For example, if both Scenario 1 and Scenario 2 occur, it means Car A is a common car in both scenarios. In this case, at a certain time, Car A would be passing Car B, Car B would be passing Car C, Car C would be passing Car A, and Car A would be passing Car D in Scenario 2.
Therefore, Car A would be passing all three other cars, meaning all four cars would be located at the same point along the track at that moment in time.
This logic can be applied to any pair of scenarios that share at least two common cars. In each case, we can conclude that all four cars would be located at the same point along the track at a given moment in time.
Hence, proving that there is a moment in time, after the start, when all four cars are located at the same point along the track.