HMMT 十一月 2011 · 冲刺赛 · 第 32 题

HMMT November 2011 — Guts Round — Problem 32

[ 17 ] Find all real numbers x satisfying 9 27 219 9 6 3 x + x + x − x + = 0 . 8 64 512

解析

[ 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 3 3 x − = x + . 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 have 8 f ( x ) > f ( y ), contradicting the fact that f is increasing, and similarly, if y < x , we have f ( x ) < f ( y ), − 1 again a contradiction. Therefore, if f ( x ) = f ( x ) and both are increasing functions in x , we require f ( 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