HMMT 二月 2012 · 几何 · 第 6 题
HMMT February 2012 — Geometry — Problem 6
题目详情
英文原题
Triangle ABC is an equilateral triangle with side length 1. Let X0, X1, . . . be an infinite sequence of
points such that the following conditions hold:
•X0is the center of ABC
•For all i≥0,X2i+1lies on segment ABandX2i+2lies on segment AC.
•For all i≥0,∡XiXi+1Xi+2= 90◦.
•For all i≥1,Xi+2lies in triangle AX iXi+1.
Find the maximum possible value of/summationtext∞
i=0|XiXi+1|, where |PQ|is the length of line segment PQ.
解析
英文解析
- 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
• For all i ≥ 0, X lies on segment AB and X lies on segment AC .0
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
√
Answer: Let Y be the foot of the perpendicular from A to X X : note that the sum we 6
0 1
wish to minimize is simply X Y + Y A . However, it is not difficult to check (for example, by AM-3
√ √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