HMMT 十一月 2015 · 冲刺赛 · 第 17 题

HMMT November 2015 — Guts Round — Problem 17

[ 10 ] Unit squares ABCD and EF GH have centers O and O respectively, and are originally situated 1 2 such that B and E are at the same position and C and H are at the same position. The squares then rotate clockwise about their centers at the rate of one revolution per hour. After 5 minutes, what is the area of the intersection of the two squares?

解析

[ 10 ] Unit squares ABCD and EF GH have centers O and O respectively, and are originally situated 1 2 such that B and E are at the same position and C and H are at the same position. The squares then rotate clockwise about their centers at the rate of one revolution per hour. After 5 minutes, what is the area of the intersection of the two squares? Proposed by: Alexander Katz √ 2 − 3 Answer: 4 Note that AE = BF = CG = DH = 1 at all times. Suppose that the squares have rotated θ radians. π π Then ∠ O O H = − θ = ∠ O DH , so ∠ HDC = − ∠ O DH = θ . Let P be the intersection 1 2 1 1 4 4 of AB and EH and Q be the intersection of BC and GH . Then P H ‖ BQ and HQ ‖ P B , and π ∠ P HG = , so P BQH - our desired intersection - is a rectangle. We have BQ = 1 − QC = 1 − sin θ 2 2 π π and HQ = 1 − cos θ , so our desired area is (1 − cos θ )(1 − sin θ ). After 5 minutes, we have θ = = , 12 6 √ 2 − 3 so our answer is . 4