HMMT 二月 1998 · CALC 赛 · 第 1 题

英文解析

1. Problem: Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walksa sinusoidal path in search of home; that is, after t minutes he is at position ( t, sin t ).
   Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. Hewalks in such a way that after he has been in the forest for m minutes, his position is ( m, cos t ).
   What is the greatest distance between Alex and Farmer Tim while they are walking in these paths?
   Solution: At arbitrary time t , Farmer Tim is at position ( t, sin t ) and Alex is at position ( t − 5 , cos t ).
   √
   Hence at time t , the distance, d , between Tim and Alex is d = (sin t − cos t ) + 25. To find the maximum 2
   value of d , we solve for t such that = 0.dddt
   (sin t − cos t )(cos t +sin t )
   dd dd 2 2
   2 2
   √
   = . Then = 0 ⇒ sin t − cos t = 0 ⇒ sin t = cos t . Equality happens if t isdt dt
   (sin t − cos t ) +252
   any constant multiple of .π
   Notice that to maximize d , we need to maximize (sin t − cos t ) . This is achieved when cos t = − sin t .24
   Because we determined earlier that t is a constant multiple of , then with this new condition, we see thatπ
   3 π4
   t must be a constant multiple of .
   √4
   Then (sin t − cos t ) = 2 ⇒ d = 29 .2