HMMT 十一月 2018 · THM 赛 · 第 9 题

HMMT November 2018 — THM Round — Problem 9

Circle ω of radius 1 and circle ω of radius 2 are concentric. Godzilla inscribes square CASH in ω 1 2 1 and regular pentagon M ON EY in ω . It then writes down all 20 (not necessarily distinct) distances 2 between a vertex of CASH and a vertex of M ON EY and multiplies them all together. What is the maximum possible value of his result?

解析

Circle ω of radius 1 and circle ω of radius 2 are concentric. Godzilla inscribes square CASH in ω 1 2 1 and regular pentagon M ON EY in ω . It then writes down all 20 (not necessarily distinct) distances 2 between a vertex of CASH and a vertex of M ON EY and multiplies them all together. What is the maximum possible value of his result? Proposed by: Michael Ren 20 Answer: 1048577 or 2 + 1 We represent the vertices with complex numbers. Place the vertices of CASH at 1 , i, − 1 , − i and the 2 πi 2 2 3 4 5 vertices of M ON EY at 2 α, 2 αω, 2 αω , 2 αω , 2 αω , 2 αω with | α | = 1 and ω = e . We have that the 4 product of distances from a point z to the vertices of CASH is | ( z − 1)( z − i )( z + 1)( z + i ) | = | z − 1 | , 4 4 4 4 3 4 2 4 so we want to maximize | (16 α − 1)(16 α ω − 1)(16 α ω − 1)(16 α ω − 1)(16 α ω − 1) | , which just 20 20 20 comes out to be | 2 α − 1 | . By the triangle inequality, this is at most 2 + 1, and it is clear that some α makes equality hold.