PUMaC 2022 · 团队赛 · 第 10 题

PUMaC 2022 — Team Round — Problem 10

Let α, β, γ ∈ C be the roots of the polynomial x − 3 x + 3 x + 7 . For any complex number z, let f ( z ) be defined as follows: f ( z ) = | z − α | + | z − β | + | z − γ | − 2 max | z − w | . w ∈{ α,β,γ } Let A be the area of the region bounded by the locus of all z ∈ C at which f ( z ) attains its global minimum. Find ⌊ A ⌋ .

解析

Let α, β, γ ∈ C be the roots of the polynomial x − 3 x + 3 x + 7 . For any complex number z, let f ( z ) be defined as follows: f ( z ) = | z − α | + | z − β | + | z − γ | − 2 max | z − w | . w ∈{ α,β,γ } Let A be the area of the region bounded by the locus of all z ∈ C at which f ( z ) attains its global minimum. Find ⌊ A ⌋ . Proposed by Oliver Thakar Answer: 12 √ The roots α, β, and γ are − 1 , 2 ± 3 i, which form an equilateral triangle in the complex plane. f ( z ) is simply the sum of the smaller two of the three distances between z and the vertices of 5 this triangle minus the largest of the distances. Ptolemy’s inequality tells us that f ( z ) ≥ 0 and it equals zero only when z lies on the circumcircle of the triangle with vertices α, β, γ ; clearly, the circumcenter of this triangle is at z = 1 , so the circumradius is 2. The area of the circle is 2 π · 2 , which has floor 12.