PUMaC 2022 · 几何（B 组） · 第 5 题

PUMaC 2022 — Geometry (Division B) — Problem 5

Daeun draws a unit circle centered at the origin and inscribes within it a regular hexagon ABCDEF . Then Dylan chooses a point P within the circle of radius 2 centered at the origin. Let M be the maximum possible value of | P A | · | P B | · | P C | · | P D | · | P E | · | P F | , and let N be 2 the number of possible points P for which this maximal value is obtained. Find M + N .

解析

Daeun draws a unit circle centered at the origin and inscribes within it a regular hexagon ABCDEF . Then Dylan chooses a point P within the circle of radius 2 centered at the origin. Let M be the maximum possible value of | P A | · | P B | · | P C | · | P D | · | P E | · | P F | , and let N be 2 the number of possible points P for which this maximal value is obtained. Find M + N . Proposed by Dylan Epstein-Gross Answer: 101 Using roots of unity, the product of lengths is 2 5 6 | z − 1 || z − a || z − a | · · · | z − a | = | z − 1 | 6 This is maximized when z = − 64, which has six solutions with M = 65. Thus the answer is 2 65 + 6 = 101.