HMMT 二月 2008 · 冲刺赛 · 第 21 题

英文解析

21. [ 10 ] Let ABC be a triangle with AB = 5, BC = 4 and AC = 3. Let P and Q be squares inside ABCwith disjoint interiors such that they both have one side lying on AB . Also, the two squares each havean edge lying on a common line perpendicular to AB , and P has one vertex on AC and Q has onevertex on BC . Determine the minimum value of the sum of the areas of the two squares.
    A BPQC
    Answer: Let the side lengths of P and Q be a and b , respectively. Label two of the vertices of 144
    P as D and E so that D lies on AB and E lies on AC , and so that DE is perpendicular to AB . The 49
    triangle ADE is similar to ACB . So AD = a . Using similar arguments, we find that 3
    3 a 4 b 4 + a + b + = AB = 5
    4 3
    a b 5 so + = .
    4 3 7
    Using Cauchy-Schwarz inequality, we get
    ( ) ( )
    ( )2
    1 1 a b 25
    2 2
    a + b + ≥ + = .
    2 2
    4 3 4 3 49
    It follows that 5
    2 2144
    a + b ≥ .
    36 4849
    Equality occurs at a = and b = .
    35 35
    11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUNDth 2