HMMT 二月 2020 · 冲刺赛 · 第 10 题

HMMT February 2020 — Guts Round — Problem 10

[6] Jarris is a weighted tetrahedral die with faces F , F , F , F . He tosses himself onto a table, so that 1 2 3 4 the probability he lands on a given face is proportional to the area of that face (i.e. the probability he [ F ] i lands on face F is where [ K ] is the area of K ). Let k be the maximum distance any i [ F ]+[ F ]+[ F ]+[ F ] 1 2 3 4 part of Jarris is from the table after he rolls himself. Given that Jarris has an inscribed sphere of radius 3 and circumscribed sphere of radius 10, find the minimum possible value of the expected value of k . 2 2

解析

[6] Jarris is a weighted tetrahedral die with faces F , F , F , F . He tosses himself onto a table, so that 1 2 3 4 the probability he lands on a given face is proportional to the area of that face (i.e. the probability [ F ] i he lands on face F is where [ K ] is the area of K ). Let k be the maximum distance i [ F ]+[ F ]+[ F ]+[ F ] 1 2 3 4 any part of Jarris is from the table after he rolls himself. Given that Jarris has an inscribed sphere of radius 3 and circumscribed sphere of radius 10, find the minimum possible value of the expected value of k . Proposed by: James Lin Answer: 12 Solution: Since the maximum distance to the table is just the height, the expected value is equal to 4 ∑ ( ) h [ F ] i i 4 ∑ i =1 1 r . Let V be the volume of Jarris. Recall that V = h [ F ] for any i , but also V = [ F ] i i i 4 ∑ 3 3 [ F ] i =1 i i =1 where r is the inradius (by decomposing into four tetrahedra with a vertex at the incenter). Therefore 4 ∑ h [ F ] i i 12 V i =1 = = 4 r = 12 . 4 ∑ 3 V /r [ F ] i i =1 2