HMMT 二月 2021 · 几何 · 第 1 题

HMMT February 2021 — Geometry — Problem 1

A circle contains the points (0 , 11) and (0 , − 11) on its circumference and contains all points ( x, y ) with 2 2 x + y < 1 in its interior. Compute the largest possible radius of the circle.

解析

A circle contains the points (0 , 11) and (0 , − 11) on its circumference and contains all points ( x, y ) with 2 2 x + y < 1 in its interior. Compute the largest possible radius of the circle. Proposed by: Carl Schildkraut Answer: 61 Solution: Such a circle will be centered at ( t, 0) for some t ; without loss of generality, let t > 0. Our conditions are that 2 2 2 t + 11 = r and r ≥ t + 1 . 2 2 So, t ≤ ( r − 1) , which means 2 2 2 ( r − 1) + 11 ≥ r = ⇒ 122 ≥ 2 r, so our answer is 61 ( r = 61 is attainable with t = 60).