Question

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40\%,$ and $m = a - b\sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c.$

答案

Solution:

Let the arrival times be $x$ and $y$ minutes after 9 a.m., so $(x, y)$ is uniform in a $60 \times 60$ square. The two people meet exactly when $|x - y| \lt m.$

The non-meeting region $|x - y| \ge m$ consists of two right triangles with legs $60 - m,$ with total area $(60 - m)^2.$ Meeting with probability $40\%$ means $(60 - m)^2 = 0.6 \cdot 3600 = 2160,$ so $60 - m = \sqrt{2160} = 12\sqrt{15}.$

Thus $m = 60 - 12\sqrt{15},$ and $a + b + c = 60 + 12 + 15 = 87.$

1998 AIME Problem 9

Solution:

Problem 9 in Other Years