HMMT 二月 2017 · 冲刺赛 · 第 11 题

HMMT February 2017 — Guts Round — Problem 11

[ 7 ] Consider the graph in 3-space of 0 = xyz ( x + y )( y + z )( z + x )( x − y )( y − z )( z − x ) . This graph divides 3-space into N connected regions. What is N ?

解析

[ 7 ] Consider the graph in 3-space of 0 = xyz ( x + y )( y + z )( z + x )( x − y )( y − z )( z − x ) . This graph divides 3-space into N connected regions. What is N ? Proposed by: Evan Chen Answer: 48 Note that reflecting for each choice of sign for x, y, z , we get new regions. Therefore, we can restrict to the case where x, y, z > 0 . In this case, the sign of the expression only depends on ( x − y )( y − z )( z − x ) . It is easy to see that for this expression, every one of the 3! = 6 orderings for { x, y, z } contributes a region. 3 Therefore, our answer is 2 · 3! = 48 .