HMMT 二月 2000 · ORAL 赛 · 第 7 题

HMMT February 2000 — ORAL Round — Problem 7

[45] A regular tetrahedron of volume 1 is filled with water of total volume 7/16. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume 1? 2

解析

If the tetrahedron is oriented so that two opposite edges of the tetrahedron are parallel to the ground, it is clear from symmetry that the plane parallel to the ground and passing through the center of the tetrahedron splits it into equal spaces of volume 1 / 2. Let’s call this position 1. If, instead, it is oriented with the point down and the top face parallel to the ground, the plane through the center splits it into regions of volume 27/64 and 37/64. To see this, note that the center of a regular tetrahedron is 3/4 of the way down the altitude from each vertex. Hence the tetrahedral section has 3 volume (3 / 4) = 27 / 64. Let’s call this position position 2. If the tetrahedron is turned continuously from position 1 to position 2, because 27 / 64 < 7 / 16 < 1 / 2, it must pass through an orientation s.t. the fraction of volume below the center is 7 / 16. ◦ The cube has 180 rotational symmetry, so any plane through the center splits it into sections of equal volume. Hence it is impossible to get a volume of 7/16. 2