HMMT 二月 2005 · 冲刺赛 · 第 39 题

英文解析

39. How many regions of the plane are bounded by the graph of
    6 5 4 2 3 2 2 4 4 6
    x − x + 3 x y + 10 x y + 3 x y − 5 xy + y = 0?
    Solution: 5
    The left-hand side decomposes as
    6 4 2 2 4 6 5 3 2 4 2 2 3 5 3 2 4
    ( x + 3 x y + 3 x y + y ) − ( x − 10 x y + 5 xy ) = ( x + y ) − ( x − 10 x y + 5 xy ) .
    Now, note that
    5 5 4 3 2 2 3 4 5
    ( x + iy ) = x + 5 ix y − 10 x y − 10 ix y + 5 xy + iy ,
    2 2 3 5
    so that our function is just ( x + y ) − < (( x + iy ) ). Switching to polar coordinates,
    6 5 5 6 5
    this is r − < ( r (cos θ + i sin θ ) ) = r − r cos 5 θ by de Moivre’s rule. The graph of our
    6 5
    function is then the graph of r − r cos 5 θ = 0, or, more suitably, of r = cos 5 θ . Thisis a five-petal rose, so the answer is 5.
    15