Thirty points are placed equally spaced on a circle, and the points are labeled with the integers from 1 to 30 in random order.

Question

Prove that no matter what the arrangement of labels is, there are three successive integers whose sum is at least 45.

Answer 1

No matter the random order, there are 30 triplets, with each number 1-30 represented in 3 triplets.

Thus, the average value of the triplets = 3 * the average of 1-30 = 3 * 15.5 = 46.5

If all triplets are less than 45, the average must be less than 45. Yet, the average is 46.5 >= 45. This is a contradiction. Thus, there must be one set of 3 consecutive integers whose sum is at least 45. In fact, there must be one set whose sum is at least 47.