HMMT 二月 2010 · 冲刺赛 · 第 20 题

HMMT February 2010 — Guts Round — Problem 20

[ 10 ] Find the volume of the set of points ( x, y, z ) satisfying x, y, z ≥ 0 x + y ≤ 1 y + z ≤ 1 z + x ≤ 1

解析

[ 10 ] Find the volume of the set of points ( x, y, z ) satisfying x, y, z ≥ 0 x + y ≤ 1 y + z ≤ 1 z + x ≤ 1 1 Answer: Without loss of generality, assume that x ≥ y - half the volume of the solid is on this 4 1 side of the plane x = y . For each value of c from 0 to , the region of the intersection of this half of 2 1 the solid with the plane y = c is a trapezoid. The trapezoid has height 1 − 2 c and average base , so 2 1 it has an area of − c . 2 1 1 1 1 The total volume of this region is times the average area of the trapezoids, which is · = . Double 2 2 4 8 1 that to get the total volume, which is . 4