HMMT 十一月 2022 · 冲刺赛 · 第 3 题

HMMT November 2022 — Guts Round — Problem 3

[5] A polygon P is drawn on the 2D coordinate plane. Each side of P is either parallel to the x axis or the y axis (the vertices of P do not have to be lattice points). Given that the interior of P includes the 2 2 interior of the circle x + y = 2022 , find the minimum possible perimeter of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2022, November 12, 2022 — GUTS ROUND Organization Team Team ID#

解析

[5] A polygon P is drawn on the 2D coordinate plane. Each side of P is either parallel to the x axis or the y axis (the vertices of P do not have to be lattice points). Given that the interior of P includes 2 2 the interior of the circle x + y = 2022 , find the minimum possible perimeter of P . Proposed by: Carl Schildkraut √ Answer: 8 2022 Solution: The minimum possible perimeter is achieved by an axis-aligned square with all four sides √ tangent to the circle, which has area 8 2022. To see why this is true, notice that there must be at √ √ √ least 2 2022 length of total perimeter facing left, 2 2022 length facing up, 2 2022 facing right, and √ √ 2 2022 facing down in order for the polygon to be closed and have a shadow of length at least 2 2022 in both the x and y directions.