HMMT 二月 2002 · 冲刺赛 · 第 36 题

英文解析

36. Find the set consisting of all real values of x such that the three numbers 2 , 2 , 2
    form a non-constant arithmetic progression (in that order).
    Solution: The empty set, Ø . Trivially, x = 0 , 1 yield constant arithmetic progressions;
    we show that there are no other possibilities. If these numbers do form a progression, then,
    by the AM-GM (arithmetic mean-geometric mean) inequality,
    √
    2 3
    x x x 3
    x x
    2 · 2 = 2 + 2 ≥ 2 2 · 2
    2 3
    x ( x + x ) / 2 2 3
    ⇒ 2 ≥ 2 ⇒ x ≥ ( x + x ) / 2
    2 3 2
    ⇒ x ( x − 1) = x − 2 x + x ≤ 0 .
    2 x x 3
    Assuming x 6 = 0 , 1, we can divide by ( x − 1) > 0 and obtain x < 0. However, then 2 , 2 arex 2
    less than 1, while 2 is more than 1, so the given sequence cannot possibly be an arithmeticprogression.