HMMT 二月 2021 · 几何 · 第 7 题

HMMT February 2021 — Geometry — Problem 7

Let O and A be two points in the plane with OA = 30, and let Γ be a circle with center O and radius ◦ r . Suppose that there exist two points B and C on Γ with ∠ ABC = 90 and AB = BC . Compute the minimum possible value of b r c .

解析

Let O and A be two points in the plane with OA = 30, and let Γ be a circle with center O and radius ◦ r . Suppose that there exist two points B and C on Γ with ∠ ABC = 90 and AB = BC . Compute the minimum possible value of b r c . Proposed by: Hahn Lheem, Milan Haiman Answer: 12 ◦ Solution: Let f denote a 45 counterclockwise rotation about point A followed by a dilation centered 1 √ ◦ A with scale factor 1 / 2. Similarly, let f denote a 45 clockwise rotation about point A followed 2 √ by a dilation centered A with scale factor 1 / 2. For any point B in the plane, there exists a point ◦ C on Γ such that ∠ ABC = 90 and AB = BC if and only if B lies on f (Γ) or f (Γ). Thus, such 1 2 points B and C on Γ exist if and only if Γ intersects f (Γ) or f (Γ). So, the minimum possible value 1 2 √ √ of r occurs when Γ is tangent to f (Γ) and f (Γ). This happens when r/ 2 + r = 30 / 2, i.e., when 1 2 √ √ 30 √ r = = 30 2 − 30. Therefore, the minimum possible value of b r c is b 30 2 − 30 c = 12. 2+1