HMMT 二月 2018 · 团队赛 · 第 5 题

HMMT February 2018 — Team Round — Problem 5

[ 30 ] Is it possible for the projection of the set of points ( x, y, z ) with 0 ≤ x, y, z ≤ 1 onto some two-dimensional plane to be a simple convex pentagon?

解析

[ 30 ] Is it possible for the projection of the set of points ( x, y, z ) with 0 ≤ x, y, z ≤ 1 onto some two-dimensional plane to be a simple convex pentagon? Proposed by: Yuan Yao 1 1 1 It is not possible. Consider P , the projection of ( , , ) onto the plane. Since for any point ( x, y, z ) 2 2 2 in the cube, (1 − x, 1 − y, 1 − z ) is also in the cube, and the midpoint of their projections will be the projection of their midpoint, which is P , the projection of the cube onto this plane will be a centrally symmetric region around P , and thus cannot be a pentagon.