HMMT 十一月 2012 · GEN 赛 · 第 10 题
HMMT November 2012 — GEN Round — Problem 10
题目详情
英文原题
[8] Let αandβbe reals. Find the least possible value of
(2 cos α+ 5 sin β−8)2+ (2 sin α+ 5 cos β−15)2.
HMMT November 2012
Saturday 10 November 2012
General Test
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解析
英文解析
- [ 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 thedistance from this point to X = (8 , 15). Thus, the closest distance from such a point to X is whenthe point is 7 units away from the origin along the segment from the origin to X . Thus, since X is 17
units away from the origin, the minimum is 10 = 100.2
General Test