HMMT 二月 2006 · 冲刺赛 · 第 22 题

HMMT February 2006 — Guts Round — Problem 22

[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

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 of f ( x )? Answer: 6 Solution: The complex roots of the polynomial must come in pairs, c and c , both of i i which have the same absolute value. If n is the number of distinct absolute values | c | i corresponding to those of non-real roots, then there are at least 2 n non-real roots of f ( 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 all i correspond 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.