HMMT 二月 2012 · 冲刺赛 · 第 8 题

英文解析

8. [ 3 ] Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eatenuntil all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating acarrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per minute and Beneats 60 carrots per minute and 30 muffins per minute, what is the minimum number of minutes it willtake them to finish the food?
   Answer: 23 . 5 or 47 / 2 Amy and Ben will continuously eat carrots, then stop (not necessarily at the same time), and continuously eat muffins until no food is left. Suppose that Amy and Ben finisheating the carrots in T minutes and the muffins T minutes later; we wish to find the minimum value
   1 2
   of T + T . Furthermore, suppose Amy finishes eating the carrots at time a , and Ben does so at time
   1 2 1
   b , so that T = max( a , b ).
   1 1 1 1
   First, suppose that a ≤ b , and let b − a = c . We have 40( T − c ) + 60 T = 1000, so T is minimized
   1 1 1 1 1 1 1
   when c = 0. Also, 30( T − 5) + 70( T − max(5 − c, 0)) = 1000. Wee see that T + T is minimized when
   2 2 1 2
   c = 5, and T + T = 23 . 5. In a similar way, we see that when b ≤ a , T + T > 23 . 5, so our answer
   1 2 1 1 1 2
   is 23.5.