HMMT 二月 2008 · 冲刺赛 · 第 9 题
HMMT February 2008 — Guts Round — Problem 9
题目详情
- [ 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 coneis 4 times the volume of the tetrahedron ABCL . Find the value of BL/LV .
11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUNDth
英文原题
[ 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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 th HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND
解析
英文解析
- [ 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 coneis 4 times the volume of the tetrahedron ABCL . Find the value of BL/LV .
Answer: Let R be the radius of the base, H the height of the cone, h the height of the pyramidπ
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 πh = H . We are given that [cone] = 4[ ABCL ], so x/y = .
x + y 4 − π
11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUNDth