PUMaC 2009 · 几何（A 组） · 第 5 题

PUMaC 2009 — Geometry (Division A) — Problem 5

Lines l and m are perpendicular. Line l partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto m into two line segments of length a 1000 a and b respectively. Determine the maximum value of b c . (The floor notation b x c denotes b largest integer not exceeding x )

解析

Lines l and m are perpendicular. Line l partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto m into two line segments of length a 1000 a and b respectively. Determine the maximum value of b c . (The floor notation b x c denotes b largest integer not exceeding x ) √ Solution. 2414. The greatest possible value of the ratio is (1 + 2). Let A and B be vertices of the convex polygon on different sides of l so that their distance from l is maximal on each side. Let K and L be the intersections of l with the sides of the polygon. Define the points K and L on extension of AK , AL respectively such that K L is parallel to l . Since the 1 1 1 1 polygon is convex, the part of the polygon on A ’s side contains AKL , and the part of the polygon on B ’s side is contained in K KLL . Therefore [ K KLL ] ≥ [ AKL ]. Let M , N be 1 1 1 1 foot of perpendicular from A , B to line l respectively, then [ AKL ] 1 AM 1 √ ≤ = ⇒ ≤ [ AK L ] 2 AM + BN 1 1 2 √ AM 1 = ⇒ ≤ √ = 1 + 2 BN 2 − 1 Equality is obtained when the polygon is a triangle with l parallel to one side.