HMMT 十一月 2019 · THM 赛 · 第 3 题

HMMT November 2019 — THM Round — Problem 3

For breakfast, Milan is eating a piece of toast shaped like an equilateral triangle. On the piece of toast rests a single sesame seed that is one inch away from one side, two inches away from another side, and four inches away from the third side. He places a circular piece of cheese on top of the toast that is tangent to each side of the triangle. What is the area of this piece of cheese?

解析

For breakfast, Milan is eating a piece of toast shaped like an equilateral triangle. On the piece of toast rests a single sesame seed that is one inch away from one side, two inches away from another side, and four inches away from the third side. He places a circular piece of cheese on top of the toast that is tangent to each side of the triangle. What is the area of this piece of cheese? Proposed by: Carl Schildkraut 49 π Answer: 9 Suppose the toast has side length s . If we draw the three line segments from the sesame seed to the s three vertices of the triangle, we partition the triangle into three smaller triangles, with areas , s , and 2 7 s 2 s , so the entire piece of toast has area . Suppose the cheese has radius r . We similarly see that the 2 ( ) 2 3 rs 7 7 49 π toast has area . Equating these, we see that r = , so the area of the cheese is π = . 2 3 3 9