HMMT 二月 2005 · CALC 赛 · 第 7 题

英文解析

7. Two ants, one starting at ( − 1 , 1), the other at (1 , 1), walk to the right along the parabola y = x such that their midpoint moves along the line y = 1 with constant 2
   speed 1. When the left ant first hits the line y = , what is its speed?1
   √2
   Solution: 3 3 − 3
   1 3
   When the left ant first hits the line y = , the right ant hits the line y = . The
   2 2
   √ √
   2 1 6 3
   left ant is then at ( − , ), and the right ant is at ( , ). Let the left ant have
   2 2 2 2
   velocity with components v and v , the right ant velocity with components w andx y x
   √ √
   v wdyy yw . Since = 2 x , = − 2 and = 6. Since the midpoint of the ants movesdx v wyx x
   1 1
   at speed 1 along the line y = 1, ( v + w ) = 1 and ( v + w ) = 0. Therefore,
   x x y y
   2 2
   √
   √ √ √ √
   2 6
   √ √
   2 v = − v = w = 6 w = 6(2 − v ). Solving for v gives = 3 − 3. Thenx y y x x x
   6+ 2
   the speed of the left ant is
   √
   √
   √ √ √
   2 2 2 2
   v + v = v + ( − 2 v ) = 3 | v | = 3 3 − 3 .
   x xx y x