HMMT 二月 2005 · 冲刺赛 · 第 2 题
HMMT February 2005 — Guts Round — Problem 2
题目详情
英文原题
- [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
英文解析
- If the two vertices are separated by a diagonal of a face of the cube, then thelocus of possible vertices adjacent to both of them is a circle of radius , the locus of 1
√42
possible vertices adjacent to exactly one of them is two circles of radius , and the 2
√2
locus of possible vertices adjacent to neither of them is a circle of radius . If the two 3
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.2
√ √3