PUMaC 2022 · 几何（B 组） · 第 3 题

PUMaC 2022 — Geometry (Division B) — Problem 3

Circle Γ is centered at (0 , 0) in the plane with radius 2022 3. Circle Ω is centered on the x -axis, passes through the point A = (6066 , 0), and intersects Γ orthogonally at the point mπ P = ( x, y ) with y > 0. If the length of the minor arc AP on Ω can be expressed as for n relatively prime positive integers m, n , find m + n . (Two circles intersect orthogonally at a point P if the tangent lines at P form a right angle.)

解析

Circle Γ is centered at (0 , 0) in the plane with radius 2022 3. Circle Ω is centered on the x -axis, passes through the point A = (6066 , 0), and intersects Γ orthogonally at the point mπ P = ( x, y ) with y > 0. If the length of the minor arc AP on Ω can be expressed as for n relatively prime positive integers m, n , find m + n . (Two circles are said to intersect orthogonally at a point P if the tangent lines at P form a right angle.) Proposed by Sunay Joshi Answer: 1349 √ √ Let O = (0 , 0). Let R = 2022 3 denote the radius of Γ, so that OA = R 3. Let r denote the √ 2 2 radius of Ω. Let Q denote the center of Ω. Since OP Q is a right triangle, P Q = r + R . 1 Since OA = OQ + QA , we have p √ 2 2 R + r + r = R 3 √ ◦ Solving, we find that r = R/ 3 = 2022. Therefore ∠ OQP = 60 and the minor arc AP ◦ corresponds to an interior angle of 120 . It follows that the desired arclength is given as 1 4044 π 1348 π · 2 πr = = , and our answer is 1348 + 1 = 1349. 3 3 1