HMMT 十一月 2017 · 团队赛 · 第 6 题

HMMT November 2017 — Team Round — Problem 6

[ 40 ] Consider five-dimensional Cartesian space 5 R = { ( x , x , x , x , x ) | x ∈ R } , 1 2 3 4 5 i and consider the hyperplanes with the following equations: • x = x for every 1 ≤ i < j ≤ 5; i j • x + x + x + x + x = − 1; 1 2 3 4 5 • x + x + x + x + x = 0; 1 2 3 4 5 • x + x + x + x + x = 1. 1 2 3 4 5 5 Into how many regions do these hyperplanes divide R ?

解析

[ 40 ] Consider five-dimensional Cartesian space 5 R = { ( x , x , x , x , x ) | x ∈ R } , 1 2 3 4 5 i and consider the hyperplanes with the following equations: • x = x for every 1 ≤ i < j ≤ 5; i j • x + x + x + x + x = − 1; 1 2 3 4 5 • x + x + x + x + x = 0; 1 2 3 4 5 • x + x + x + x + x = 1. 1 2 3 4 5 5 Into how many regions do these hyperplanes divide R ? Proposed by: Mehtaab Sawhney Answer: 480 (Joint with Junyao Peng) Note that given a set of plane equations P ( x , x , x , x , x ) = 0, for i = 1 , 2 , . . . , n , each region i 1 2 3 4 5 that the planes separate the space into correspond to a n -tuple of − 1 and 1, representing the sign of P , P , . . . P for all points in that region. 1 2 n Therefore, the first set of planes separate the space into 5! = 120 regions, with each region representing an ordering of the five coordinates by numerical size. Moreover, the next three planes are parallel to each other and perpendicular to all planes in the first set, so these three planes separate each region into 4. Therefore, a total of 4 · 120 = 480 regions is created.