HMMT 二月 2022 · 团队赛 · 第 6 题

6. [45] Let P ( x ) = x + ax + bx + x be a polynomial with four distinct roots that lie on a circle in the complex plane. Prove that ab ̸ = 9. Proposed by: Akash Das Answer: Solution: If either a = 0 the problem statement is clearly true. Thus, assume that a ̸ = 0. Let the roots be 0 , z , z , z , and let the circle through these points be C . Note that we have 1 2 3 3 3 = − , z + z + z a 1 2 3 1 1 1

b z z z 1 2 3 = − . 3 3 1 Note that the map z → maps C to some line L . Thus, the second equation represents the average z of three points on L , which must be a point on L , while the second equation represents the reciprocal of the centroid of z , z , z . Since this centroid doesn’t lie on C, we must have its reciprocal doesn’t lie 1 2 3 on L . Thus, we have 3 b − ̸ = − = ⇒ ab ̸ = 9 . a 3