HMMT 二月 2007 · 冲刺赛 · 第 20 题

英文解析

20. [ 10 ] For a a positive real number, let x , x , x be the roots of the equation x − ax + ax − a = 0.
    1 2 3
    3 3 3
    Determine the smallest possible value of x + x + x − 3 x x x .
    1 2 3
    1 2 3
    Answer: − 4 . Note that x + x + x = x x + x x + x x = a . Then
    1 2 3 1 2 2 3 3 1
    3 3 3 2 2 2
    x + x + x − 3 x x x = ( x + x + x )( x + x + x − ( x x + x x + x x ))
    1 2 3 1 2 3 1 2 2 3 3 1
    1 2 3 1 2 3
    ( )
    2 3 22 = ( x + x + x ) ( x + x + x ) − 3 ( x x + x x + x x ) = a · ( a − 3 a ) = a − 3 a .
    1 2 3 1 2 3 1 2 2 3 3 1
    The expression is negative only where 0 < a < 3, so we need only consider these values of a . Finally,
    √
    (6 − 2 a )+ a + a
    AM-GM gives (6 − 2 a )( a )( a ) ≤ = 2, with equality where a = 2, and this rewrites as 3
    ( a − 3) a ≥ − 4.23