HMMT 二月 2003 · 冲刺赛 · 第 28 题

HMMT February 2003 — Guts Round — Problem 28

[10] A point in three-space has distances 2 , 6 , 7 , 8 , 9 from five of the vertices of a regular octahedron. What is its distance from the sixth vertex?

解析

A point in three-space has distances 2 , 6 , 7 , 8 , 9 from five of the vertices of a regular octahedron. What is its distance from the sixth vertex? √ Solution: 21 By a simple variant of the British Flag Theorem, if ABCD is a square and P any point 2 2 2 2 in space, AP + CP = BP + DP . Four of the five given vertices must form a square ABCD , and by experimentation we find their distances to the given point P must be AP = 2 , BP = 6 , CP = 9 , DP = 7. Then A, C , and the other two vertices E, F also √ 2 2 2 2 2 2 form a square AECF , so 85 = AP + CP = EP + F P = 8 + F P ⇒ F P = 21.