HMMT 二月 2006 · GEN1 赛 · 第 5 题

英文解析

5. In the plane, what is the length of the shortest path from ( − 2 , 0) to (2 , 0) that avoidsthe interior of the unit circle (i.e., circle of radius 1) centered at the origin?
   √
   Answer: 2 3 +π
   Solution: The path goes in a line segment tangent to the circle, then an arc of the 3
   circle, then another line segment tangent to the circle. Since one of these tangent linesand 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.