HMMT 十一月 2023 · 团队赛 · 第 5 题

HMMT November 2023 — Team Round — Problem 5

[35] A complex quartic polynomial Q is quirky if it has four distinct roots, one of which is the sum of the 4 3 2 other three. There are four complex values of k for which the polynomial Q ( x ) = x − kx − x − x − 45 is quirky. Compute the product of these four values of k .

解析

[35] A complex quartic polynomial Q is quirky if it has four distinct roots, one of which is the sum of the 4 3 2 other three. There are four complex values of k for which the polynomial Q ( x ) = x − kx − x − x − 45 is quirky. Compute the product of these four values of k . Proposed by: Pitchayut Saengrungkongka Answer: 720 Solution: Let the roots be a, b, c, d with a + b + c = d . Since a + b + c = k − d by Vieta’s formulas, we have d = k/ 2. Hence 4 3 2 4 2 k k k k k k k k 0 = P = − k − − − 45 = − − − − 45 . 2 2 2 2 2 16 4 2 We are told that there are four distinct possible values of k , which are exactly the four solutions to the above equation; by Vieta’s formulas, their product 45 · 16 = 720 .