Bicycle Path

Book: A Walk through Combinatorics

Problem

A bicycle path is 30 miles long, with four rest areas. Prove there either there are two rest areas that are at most six miles from each other, or there is a rest area that is at most six miles away from the endpoints of the path.

Solution

Let the rest areas divide the path as follows:

|----|-----|-----|-----|-----|
  a_1  a_2   a_3   a_4    a_5

We know that

We need to prove that there is some $i$ so that, for some $i$, $a_i \leq 6$.

By contradiction, if we assume that for all $i$, $a_i \gt 6$, then

which is a contradiction, we know the sum is equal to $30$. So our initial assumption is wrong, there must be some $i$ so that $a_i \leq 6$.