HMMT 十一月 2024 · 冲刺赛 · 第 26 题

HMMT November 2024 — Guts Round — Problem 26

[13] A right rectangular prism of silly powder has dimensions 20 × 24 × 25 . Jerry the wizard applies 10 bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by 1 and decreases a different dimension of the silly powder by 1 , with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine.

解析

[13] A right rectangular prism of silly powder has dimensions 20 × 24 × 25. Jerry the wizard applies 10 bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by 1 and decreases a different dimension of the silly powder by 1, with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine. Proposed by: Aaron Guo Answer: 11770 Solution: Consider the expected change in volume by one bout of highdroxylation. Let the initial dimensions of the silly powder be a , b , and c . Should a increase to a + 1 and b decrease to b − 1, the volume of the silly powder will change from abc to ( a + 1)( b − 1) c = abc + bc − ac − c. If a decreases to a − 1 and b increases to b + 1 instead, the new volume is ( a − 1)( b + 1) c = abc − bc + ac − c. Thus, the average change in volume between these two cases is − c . By symmetry, the expected change a + b + c in volume from one bout is δ = − . 3 Initially, the dimensions sum to 69, and this sum can never change. Hence, for each bout of hydrox- 69 ylation, the expected change in volume is δ = − = − 23. By linearity of expectation, the expected 3 volume after 10 bouts is 20 · 24 · 25 − 10 · 23 = 11770 .