HMMT 二月 2008 · 冲刺赛 · 第 9 题

HMMT February 2008 — Guts Round — Problem 9

[ 6 ] Consider a circular cone with vertex V , and let ABC be a triangle inscribed in the base of the cone, such that AB is a diameter and AC = BC . Let L be a point on BV such that the volume of the cone is 4 times the volume of the tetrahedron ABCL . Find the value of BL/LV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND

解析

[ 6 ] Consider a circular cone with vertex V , and let ABC be a triangle inscribed in the base of the cone, such that AB is a diameter and AC = BC . Let L be a point on BV such that the volume of the cone is 4 times the volume of the tetrahedron ABCL . Find the value of BL/LV . π Answer: Let R be the radius of the base, H the height of the cone, h the height of the pyramid 4 − π 1 1 2 2 and let BL/LV = x/y . Let [ · ] denote volume. Then [cone] = πR H and [ ABCL ] = πR h and 3 3 x π h = H . We are given that [cone] = 4[ ABCL ], so x/y = . x + y 4 − π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND