HMMT 二月 2012 · 冲刺赛 · 第 8 题

HMMT February 2012 — Guts Round — Problem 8

[ 3 ] Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per minute and Ben eats 60 carrots per minute and 30 muffins per minute, what is the minimum number of minutes it will take them to finish the food? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TH 15 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 11 FEBRUARY 2012 — GUTS ROUND

解析

[ 3 ] Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per minute and Ben eats 60 carrots per minute and 30 muffins per minute, what is the minimum number of minutes it will take 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 finish eating 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.