HMMT 二月 2024 · 几何 · 第 1 题
HMMT February 2024 — Geometry — Problem 1
题目详情
- Inside an equilateral triangle of side length 6 , three congruent equilateral triangles of side length x with sides parallel to the original equilateral triangle are arranged so that each has a vertex on a side of the larger triangle, and a vertex on another one of the three equilateral triangles, as shown below. 1 6 A smaller equilateral triangle formed between the three congruent equilateral triangles has side length
解析
1 . Compute x . Proposed by: Rishabh Das 5 Answer: 3 Solution: O √ 3 6 √ √ 3 3 x 2 Let x be the side length of the shaded triangles. Note that the centers of the triangles with side lengths 1 and 6 coincide; call this common center O . √ The distance from O to a side of the equilateral triangle with side length 1 is 3 / 6 . Similarly the √ distance from O to a side of the equilateral triangle with side length 6 is 3 . Notice the difference of these two distances is exactly the length of the altitude of one of shaded triangles. So √ √ √ 3 3 5 3 − = x = ⇒ x = . 6 2 3 ◦