Five Points, Ten Distances

Five points, A, B, C, D, and E, lie on a line. The distances between them in ascending order are: 2, 5, 6, 8, 9, X, 15, 17, 20, and 22. What is X?

We assume that the points are ordered A to E from left to right. We have AE = 22 and either AD = 20, BE = 17, or AD = 17, BE = 20. Without loss of generality AD = 20, BE = 17, and therefore AB = 5, BD = 15, DE = 2. The distance of 6 is associated with either BC or CD, and therefore the points are arranged in one of these two ways:

  1. AB = 5, BC = 6, CD = 9, DE = 2
  2. AB = 5, BC = 9, CD = 6, DE = 2

If it is the latter, we get the sequence of distances: 2, 5, 6, 9, 11, 14, 15, 17, 20, 22, which does not fit the provided sequence.

If it is the former, we get the sequence of distances: 2, 5, 6, 8, 9, 14, 15, 17, 20, 22, and therefore X = 14.

Cut the Pizza

Cut a circular pizza into 12 congruent slices, such that exactly half of them contain crust.

Remark: We say that a slice contains crust if it shares an arc with the boundary of the pizza (with non-zero measure).

First, cut the pizza into 6 congruent circular triangles, and then split each of them in half, as shown on the image below.