HMMT 十一月 2012 · GEN 赛 · 第 10 题

HMMT November 2012 — GEN Round — Problem 10

[ 8 ] Let α and β be reals. Find the least possible value of 2 2 (2 cos α + 5 sin β − 8) + (2 sin α + 5 cos β − 15) . HMMT November 2012 Saturday 10 November 2012 General Test Name Team ID# School Team

解析

[ 8 ] Let α and β be reals. Find the least possible value of 2 2 (2 cos α + 5 sin β − 8) + (2 sin α + 5 cos β − 15) . − → − → Answer: 100 Let the vector v = (2 cos α, 2 sin α ) and w = (5 sin β, 5 cos β ). The locus of ends of − → − → vectors expressible in the form v + w are the points which are five units away from a point on the circle of radius two about the origin. The expression that we desire to minimize is the square of the distance from this point to X = (8 , 15). Thus, the closest distance from such a point to X is when the point is 7 units away from the origin along the segment from the origin to X . Thus, since X is 17 2 units away from the origin, the minimum is 10 = 100. General Test