HMMT 二月 2021 · 冲刺赛 · 第 19 题

HMMT February 2021 — Guts Round — Problem 19

[12] Almondine has a bag with N balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger than ⌊ ⌋ N 23 percent. Compute the largest possible value of . 3 3

解析

[12] Almondine has a bag with N balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger ⌊ ⌋ N than 23 percent. Compute the largest possible value of . 3 Proposed by: James Lin Answer: 29 3 N 6 k Solution: If k = b c , then the maximum possible probability is . with equality when 3 (3 k )(3 k − 1)(3 k − 2) k k +1 there are k balls of each of the three colors. Going from 3 k → 3 k + 1 replaces → , which is 3 k − 2 3 k +1 k k +1 smaller, and going from 3 k + 1 → 3 k + 2 replaces → , which is again smaller. For this to be 3 k − 1 3 k +2 23 2 larger than , we find we need 0 > 7 k − 207 k + 46, and so k = 29 is the maximal value. 100 3