HMMT 十一月 2013 · 冲刺赛 · 第 29 题

HMMT November 2013 — Guts Round — Problem 29

[ 15 ] Let 4 XY Z be a right triangle with ∠ XY Z = 90 . Suppose there exists an infinite sequence of equilateral triangles X Y T , X Y T , . . . such that X = X, Y = Y , X lies on the segment XZ for all 0 0 0 1 1 1 0 0 i i ≥ 0, Y lies on the segment Y Z for all i ≥ 0, X Y is perpendicular to Y Z for all i ≥ 0, T and Y are i i i i separated by line XZ for all i ≥ 0, and X lies on segment Y T for i ≥ 1. i i − 1 i − 1 Let P denote the union of the equilateral triangles. If the area of P is equal to the area of XY Z , find XY . Y Z 2 2 2

解析

[ 15 ] Let 4 XY Z be a right triangle with ∠ XY Z = 90 . Suppose there exists an infinite sequence of equilateral triangles X Y T , X Y T , . . . such that X = X, Y = Y , X lies on the segment XZ for all 0 0 0 1 1 1 0 0 i i ≥ 0, Y lies on the segment Y Z for all i ≥ 0, X Y is perpendicular to Y Z for all i ≥ 0, T and Y are i i i i separated by line XZ for all i ≥ 0, and X lies on segment Y T for i ≥ 1. i i − 1 i − 1 Let P denote the union of the equilateral triangles. If the area of P is equal to the area of XY Z , find XY . Y Z Answer: 1 For any region R , let [ R ] denote its area. 2 4 Let a = XY , b = Y Z , ra = X Y . Then [ P ] = [ XY T ](1 + r + r + · · · ), [ XY Z ] = [ XY Y X ](1 + 1 1 0 1 1 √ √ 2 4 2 r + r + · · · ), Y Y = ra 3, and b = ra 3(1 + r + r + · · · ) (although we can also get this by similar 1 triangles). √ √ √ 2 a 3 1 3 − 1 XY a 1 − r √ Hence = ( ra + a )( ra 3), or 2 r ( r + 1) = 1 = ⇒ r = . Thus = = = 1. 4 2 2 Y Z b r 3 2 2 2