HMMT 二月 2012 · 几何 · 第 6 题

HMMT February 2012 — Geometry — Problem 6

Triangle ABC is an equilateral triangle with side length 1. Let X , X , . . . be an infinite sequence of 0 1 points such that the following conditions hold: • X is the center of ABC 0 • For all i ≥ 0, X lies on segment AB and X lies on segment AC . 2 i +1 2 i +2 ◦ • For all i ≥ 0, ∡ X X X = 90 . i i +1 i +2 • For all i ≥ 1, X lies in triangle AX X . i +2 i i +1 ∑ ∞ Find the maximum possible value of | X X | , where | P Q | is the length of line segment P Q . i i +1 i =0

解析

Triangle ABC is an equilateral triangle with side length 1. Let X , X , . . . be an infinite sequence of 0 1 points such that the following conditions hold: • X is the center of ABC 0 • For all i ≥ 0, X lies on segment AB and X lies on segment AC . 2 i +1 2 i +2 ◦ • For all i ≥ 0, ∡ X X X = 90 . i i +1 i +2 • For all i ≥ 1, X lies in triangle AX X . i +2 i i +1 ∑ ∞ Find the maximum possible value of | X X | , where | P Q | is the length of line segment P Q . i i +1 i =0 √ 6 Answer: Let Y be the foot of the perpendicular from A to X X : note that the sum we 0 1 3 wish to minimize is simply X Y + Y A . However, it is not difficult to check (for example, by AM- 0 √ √ ◦ GM) that AY + Y X ≥ 2 AX = 6 / 3. This may be achieved by making ∠ Y X A = 45 , so that 0 0 0 ◦ ∠ AX X = 105 . 1 0