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
ying 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.
Let's assume that each car's speed is measured in distance units per time unit. Since all four cars start at the same time and are going at constant speeds, after a certain time, they would have traveled a certain distance along the loop track.
Consider the four distances traveled by the four cars after this certain time. These distances can be represented as a, b, c, and d.
Without loss of generality, let's assume that a ≤ b ≤ c ≤ d. Since all three possible pairs of cars have a moment in time when they pass each other, we can determine the relative distances between each pair of cars.
For the pair (a, b), there is a moment when the first car (car A) is at distance a and the second car (car B) is at distance b. This means that the distance between the two cars is (b - a).
Similarly, for the pair (a, c), the distance between car A and car C is (c - a). And for the pair (b, c), the distance between car B and car C is (c - b).
Now, let's consider the differences between these distances:
1. The difference between (b - a) and (c - a) is (b - c).
2. The difference between (b - a) and (c - b) is (2b - a - c).
3. The difference between (c - a) and (c - b) is (b - a).
Since all three of these differences must be non-negative, we have:
1. (b - c) ≥ 0
2. (2b - a - c) ≥ 0
3. (b - a) ≥ 0
From (1), we know that b ≥ c.
From (3), we know that b ≥ a.
Combining these two inequalities, we get c ≤ b ≤ a.
But we initially assumed that a ≤ b ≤ c ≤ d.
Hence, c ≤ b ≤ a contradicts our assumption. Therefore, our initial assumption that each car's speed is measured in distance units per time unit must be false.
In other words, it is not possible for each car to have different constant speeds. Therefore, 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 have more pigeons than pigeonholes, then at least one pigeonhole must have more than one pigeon.
In this scenario, the four cars can be represented as pigeons, and the different points along the track can be represented as pigeonholes. Since there are infinitely many points on the loop track, there are more points than there are cars. Therefore, we have more "pigeonholes" than "pigeons."
Now, let's assume that there is no moment in time when all four cars are located at the same point along the track. This means that at any given moment, there is always at least one car located at a different point than the other three cars.
Now consider three of the cars. According to the given condition, there must be a moment in time when these three cars are located at the same point along the track. Since the fourth car is always at a different point, it cannot be part of this group.
We can repeat this process for all possible combinations of three cars. In each case, there will be a moment in time when those three cars are located at the same point along the track. However, the fourth car will always be at a different point.
Since there are four possible combinations of three cars (ABCD, ABDC, ACBD, and ACDB), and in each case, there is a moment when those three cars are at the same point, and the fourth car is at a different point, we have more pigeons (combinations) than pigeonholes (points on the track).
According to the Pigeonhole Principle, at least one pigeonhole must have more than one pigeon. This means that there must be a moment in time when at least two of the combinations overlap, i.e., when all four cars are located at the same point along the track.
Therefore, we have proved that there is a moment in time, after the start, when all four cars are located at the same point along the track.