HMMT 二月 2006 · 冲刺赛 · 第 22 题
HMMT February 2006 — Guts Round — Problem 22
题目详情
英文原题
- [9] Let f ( x ) be a degree 2006 polynomial with complex roots c , c , . . . , c , such that the set
1 2 2006
{| c | , | c | , . . . , | c |} consists of exactly 1006 distinct values. What is the minimum number
1 2 2006
of real roots of f ( x )?
解析
英文解析
- Let f ( x ) be a degree 2006 polynomial with complex roots c , c , . . . , c , such that
1 2 2006
the set
{| c | , | c | , . . . , | c |}
1 2 2006
consists of exactly 1006 distinct values. What is the minimum number of real roots off ( x )?
Answer: 6
Solution: The complex roots of the polynomial must come in pairs, c and c , both ofi iwhich have the same absolute value. If n is the number of distinct absolute values | c |
corresponding to those of non-real roots, then there are at least 2 n non-real roots ofif ( x ). Thus f ( x ) can have at most 2006 − 2 n real roots. However, it must have at least
1006 − n real roots, as | c | takes on 1006 − n more values. By definition of n , these allicorrespond to real roots. Therefore 1006 − n ≤ # real roots ≤ 2006 − 2 n , so n ≤ 1000,
and # real roots ≥ 1006 − n ≥ 6. It is easy to see that equality is attainable.