HMMT 二月 2019 · 冲刺赛 · 第 30 题

30. [ 20 ] Three points are chosen inside a unit cube uniformly and independently at random. What is the 1 probability that there exists a cube with side length and edges parallel to those of the unit cube that 2 contains all three points? Proposed by: Yuan Yao 1 Answer: 8 Let the unit cube be placed on a xyz -coordinate system, with edges parallel to the x, y, z axes. Suppose 1 the three points are labeled A, B, C. If there exists a cube with side length and edges parallel to 2 1 the edges of the unit cube that contain all three points, then there must exist a segment of length 2 that contains all three projections of A, B, C onto the x -axis. The same is true for the y - and z - axes. 1 Likewise, if there exists segments of length that contains each of the projections of A, B, C onto the 2 1 x , y , and z axes, then there must exist a unit cube of side length that contains A, B, C . It is easy 2 to see that the projection of a point onto the x -axis is uniform across a segment of length 1, and that each of the dimensions are independent. The problem is therefore equivalent to finding the cube of the 1 probability that a segment of length can cover three points chosen randomly on a segment of length 2

Note that selecting three numbers p < q < r uniformly and independently at random from 0 to 1 splits the number line into four intervals. That is, we can equivalently sample four positive numbers a, b, c, d uniformly satisfying a + b + c + d = 1 (here, we set a = p, b = q − p, c = r − q, d = 1 − r ). The probability 1 1 1 that the points p, q, r all lie on a segment of length is the probability that r − q ≤ , or b + c ≤ . 2 2 2 1 1 3 1 Since a + d and b + c are symmetric, we have that this probability is and our final answer is ( ) = . 2 2 8