HMMT 十一月 2012 · 冲刺赛 · 第 28 题

英文解析

28. [ 15 ]
    √
    1+ 2 ab + b 2
    Answer: (0 , ] Suppose that k = for some positive real a, b . We claim that k lies in
    2 2
    2 a + b
    √
    2 a +1
    1+ 2 a ab + b x +1 2
    (0 , ]. Let x = . We have that = = . Thus, x + 1 = k ( x + 1), so theb
    2 2 2 2
    2 b a + b x +1 a +1
    ( )
    2 bquadratic kt − t + k − 1 = 0 has a positive real root. Thus, its discriminant must be nonnegative, so
    √ √
    1 − 2 1+ 2
    2 2
    1 ≥ 4( k − 1)( k ) = ⇒ (2 k − 1) ≤ 2, which implies ≤ k ≤ . Since x > 0, we also have
    2 2
    √
    1+ 2
    k > 0, so we know that k must lie in (0 , ].
    √2
    1+ 2
    Now, take any k in the interval (0 , ]. We thus know that 1 ≥ 4 k ( k − 1), so the quadratic 2
    √2
    1+ 1 − 4 k ( k − 1)
    2 2
    kt − t + k − 1 = 0 has a positive solution, . Call this solution x . Then k ( x + 1) = x + 1,
    2 kx +1 ab + b 2
    so = k . If we set a = x and b = 1, we get that = k . Thus, the set of all attainable values
    2 2 2
    x +1 a + b
    √
    ab + b 1+ 22
    of is the interval (0 , ].
    2 2
    a + b 2