HMMT 二月 2005 · 冲刺赛 · 第 2 题

HMMT February 2005 — Guts Round — Problem 2

[5] Let x , y , and z be positive real numbers such that ( x · y ) + z = ( x + z ) · ( y + z ). What is the maximum possible value of xyz ?

解析

If the two vertices are separated by a diagonal of a face of the cube, then the 1 locus of possible vertices adjacent to both of them is a circle of radius , the locus of 2 4 √ 2 possible vertices adjacent to exactly one of them is two circles of radius , and the 2 √ 3 locus of possible vertices adjacent to neither of them is a circle of radius . If the two 2 given vertices are separated by a long diagonal, then each of the other vertices lie on √ 2 one of two circles of radius , for a total of 10 circles. 3 √ √