HMMT 二月 2006 · GEN1 赛 · 第 5 题

HMMT February 2006 — GEN1 Round — Problem 5

In the plane, what is the length of the shortest path from ( − 2 , 0) to (2 , 0) that avoids the interior of the unit circle (i.e., circle of radius 1) centered at the origin?

解析

In the plane, what is the length of the shortest path from ( − 2 , 0) to (2 , 0) that avoids the interior of the unit circle (i.e., circle of radius 1) centered at the origin? √ π Answer: 2 3 + 3 Solution: The path goes in a line segment tangent to the circle, then an arc of the circle, then another line segment tangent to the circle. Since one of these tangent lines and a radius of the circle give two legs of a right triangle with hypotenuse the line from √ √ 2 2 (0,0) to (-2,0) or (2,0), the length of each tangent line is 2 − 1 = 3. Also, because ◦ ◦ ◦ ◦ these are 30 -60 -90 right triangles, the angle of the arc is 60 and has length π/ 3.