HMMT 十一月 2011 · 冲刺赛 · 第 26 题

英文解析

26. [ 12 ] Determine the positive real value of x for which
    √
    √ √
    2 + AC + 2 Cx + AC − 2 + 2 Ax = 2( A + C ) x + 2 AC.
    √
    √
    √
    Answer: 4 Note that if we have a + b = a + b for non-negative reals a, b , then squaring gives
    √
    us that 2 ab = 0, so that either a = 0 or b = 0.
    Now, note that
    (2 + AC + 2 Cx ) + ( AC − 2 + 2 Ax ) = (2( A + C ) x + 2 AC ) .
    Guts Round
    Consequently, either (2 + AC + 2 Cx ) or ( AC − 2 + 2 Ax ) must be equal to 0. However, we observefrom the problems that both A , C , and x must be non-negative, so (2 + AC + 2 Cx ) > 0. As a result,
    we know that AC − 2 + 2 Ax = 0, or that
    2 − AC
    B = x = .
    2 A
    If we solve our system of equations for A, B, C , we get that B = 4.