HMMT 二月 2022 · 冲刺赛 · 第 2 题

HMMT February 2022 — Guts Round — Problem 2

[5] Three distinct vertices are randomly selected among the five vertices of a regular pentagon. Let p be the probability that the triangle formed by the chosen vertices is acute. Compute 10 p .

解析

Let this quantity be g ( n ) = a / 2. n Divide the n th iteration fractal into three meta-columns of equal width. Then the probability that a 3 2 3 random point is in the first, second, and third meta-columns is , , and , respectively. If the two 8 8 8 points end up in neighboring meta columns, the expected value of their x -distance is simply the width n − 1 of a meta-column, which is 3 . If they end up in opposite meta-columns (the left and right ones), it n − 1 is twice this amount, which is 2 · 3 . Finally, if the two points lie in the same meta-column, which 3 2 2 2 3 2 11 happens with probability ( ) + ( ) + ( ) = , the expected x -distance is just g ( n − 1). Thus, we 8 8 8 32 have 3 2 2 3 3 3 11 15 11 n − 1 n − 1 n − 1 g ( n ) = 3 2 · · + 2 · · + 2 · 3 2 · · + g ( n − 1) = · 3 + g ( n − 1) . 8 8 8 8 8 8 32 16 32 n As n grows, say this is asymptotic to g ( n ) = 3 C . For some constant C . Then we can write n 15 n − 1 11 n − 1 6 12 3 C = · 3 + · 3 C = ⇒ C = . Our final answer is twice this, which is . 16 32 17 17