HMMT 十一月 2019 · 冲刺赛 · 第 33 题

HMMT November 2019 — Guts Round — Problem 33

[17] A circle Γ with center O has radius 1. Consider pairs ( A, B ) of points so that A is inside the circle and B is on its boundary. The circumcircle Ω of OAB intersects Γ again at C 6 = B , and line AC intersects Γ again at X 6 = C . The pair ( A, B ) is called techy if line OX is tangent to Ω. Find the area of the region of points A so that there exists a B for which ( A, B ) is techy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2019, November 9, 2019 — GUTS ROUND Organization Team Team ID#

解析

[17] A circle Γ with center O has radius 1. Consider pairs ( A, B ) of points so that A is inside the circle and B is on its boundary. The circumcircle Ω of OAB intersects Γ again at C 6 = B , and line AC intersects Γ again at X 6 = C . The pair ( A, B ) is called techy if line OX is tangent to Ω. Find the area of the region of points A so that there exists a B for which ( A, B ) is techy. Proposed by: Carl Schildkraut and Milan Haiman 3 π Answer: 4 We claim that ( A, B ) is techy if and only if OA = AB . Note that OX is tangent to the circle ( OBC ) if and only if OX is perpendicular to the angle bisector of ∠ BOC , since OB = OC . Thus ( A, B ) is techy if and only if OX is parallel to BC . Now since OC = OX , OX ‖ BC ⇐⇒ ∠ BCA = ∠ OXA ⇐⇒ ∠ BCA = ∠ ACO ⇐⇒ OA = AB. From the claim, the desired region of points A is an annulus between the circles centered at O with 1 3 π radii and 1. So the answer is . 2 4