HMMT 十一月 2016 · 冲刺赛 · 第 15 题

HMMT November 2016 — Guts Round — Problem 15

[ 9 ] Let ABCD be an isosceles trapezoid with parallel bases AB = 1 and CD = 2 and height 1. Find the area of the region containing all points inside ABCD whose projections onto the four sides of the trapezoid lie on the segments formed by AB , BC , CD and DA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2016, November 12, 2016 — GUTS ROUND Organization Team Team ID#

解析

[ 9 ] Let ABCD be an isosceles trapezoid with parallel bases AB = 1 and CD = 2 and height 1. Find the area of the region containing all points inside ABCD whose projections onto the four sides of the trapezoid lie on the segments formed by AB , BC , CD and DA . Proposed by: Allen Liu 5 Answer: 8 Let E , F , be the projections of A , B on CD . A point whose projections lie on the sides must be contained in the square ABF E . Furthermore, the point must lie under the perpendicular to AD at A 1 1 and the perpendicular to BC at B , which have slopes and − . The area of the desired pentagon is 2 2 5 1 1 1 − − = . 4 8 8