HMMT 二月 2021 · 冲刺赛 · 第 6 题

HMMT February 2021 — Guts Round — Problem 6

[9] In a group of 50 children, each of the children in the group have all of their siblings in the group. Each child with no older siblings announces how many siblings they have; however, each child with an older sibling is too embarrassed, and says they have 0 siblings. 12 If the average of the numbers everyone says is , compute the number of different sets of siblings 25 represented in the group.

解析

[9] In a group of 50 children, each of the children in the group have all of their siblings in the group. Each child with no older siblings announces how many siblings they have; however, each child with an older sibling is too embarrassed, and says they have 0 siblings. 12 If the average of the numbers everyone says is , compute the number of different sets of siblings 25 represented in the group. Proposed by: Vincent Bian Answer: 26 Solution: For i ≥ 1 , let a be the number of families that have i members in the group. Then, among i each family with i children in the group, the oldest child will say i − 1 , and the rest will say 0 . Thus, 12 the sum of all the numbers said will be a + 2 a + 3 a + 4 a + · · · = 50 × = 24 . 2 3 4 5 25 Also because there are 50 children total, we know that a + 2 a + 3 a + · · · = 50 . We can subtract 1 2 3 these two equations to get a + a + a + · · · = 50 − 24 = 26 . 1 2 3