PUMaC 2015 · 几何（A 组） · 第 4 题

PUMaC 2015 — Geometry (Division A) — Problem 4

[ 4 ] Find the largest r such that 4 balls each of radius r can be packed into a regular tetrahedron with side length 1. In a packing, each ball lies outside every other ball, and every ball lies ” a + b inside the boundaries of the tetrahedron. If r can be expressed in the form where a, b, c c are integers such that gcd ( b, c ) = 1, what is a + b + c ? É

解析

[ 4 ] Find the largest r such that 4 balls each of radius r can be packed into a regular tetrahedron with side length 1. In a packing, each ball lies outside every other ball, and every ball lies ” a + b inside the boundaries of the tetrahedron. If r can be expressed in the form where a, b, c c are integers such that gcd ( b, c ) = 1, what is a + b + c ? Solution: Let the radius be r . It is obvious that the largest r is achieved when the four balls are all tangent to three faces of the regular tetrahedron, one ball at a corner. A direct calculation shows that the distance from the center of a ball to that vertex of tetrahedron is ÷ 3 3 r , and the distance from the center of a ball to the center of tetrahedron is r , and therefore 2 ÷ 3 the distance from a vertex to the center of the tetrahedron is ( 3 + ) r . So the side length of 2 ” ” 1 6 − 1 ” the tetrahedron is ( 2 6 + 2 ) r and therefore r = = , and our answer is 15 . 10 2 6 + 2 Author: Xiaoyu Xu 2 É