HMMT 十一月 2011 · 冲刺赛 · 第 32 题

英文解析

32. [ 17 ] Find all real numbers x satisfying
    9 27 219
    9 6 3
    x + x + x − x + = 0 .
    8 64 512
    √
    1 − 1 ± 13
    Answer: , Note that we can re-write the given equation as
    2 4
    √
    3 3
    x − = x + .33
    8 8
    Furthermore, the functions of x on either side, we see, are inverses of each other and increasing. Let
    √
    3 3 − 1
    f ( x ) = x − . Suppose that f ( x ) = y = f ( x ). Then, f ( y ) = x . However, if x < y , we havef ( x ) > f ( y ), contradicting the fact that f is increasing, and similarly, if y < x , we have f ( x ) < f ( y ),8 − 1
    again a contradiction. Therefore, if f ( x ) = f ( x ) and both are increasing functions in x , we requiref ( x ) = x . This gives the cubic
    ( ) ( )
    3 1 1 3
    3 2
    x − x + = 0 → x − x + x − = 0 ,
    8 2 2 4
    √
    1 − 1 ± 13
    giving x = , .
    2 4