HMMT 十一月 2015 · 冲刺赛 · 第 17 题

英文解析

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 thenrotate clockwise about their centers at the rate of one revolution per hour. After 5 minutes, what isthe area of the intersection of the two squares?
    Proposed by: Alexander Katz
    √
    2 − 3
    Answer:
    Note that AE = BF = CG = DH = 1 at all times. Suppose that the squares have rotated θ radians.4
    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 , andπ
    ∠ P HG = , so P BQH - our desired intersection - is a rectangle. We have BQ = 1 − QC = 1 − sin θ
    2 π π2
    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