PUMaC 2023 · 团队赛 · 第 9 题

PUMaC 2023 — Team Round — Problem 9

The real quartic P x + U x + M x + Ax + C has four different positive real roots. Find the 2 square of the smallest real number z for which the expression M − 2 U A + zP C is always positive, regardless of what the roots of the quartic are. 1 2020 X 4 kπ

解析

The real quartic P x + U x + M x + Ax + C has four different positive real roots. Find the 2 square of the smallest real number z for which the expression M − 2 U A + zP C is always positive, regardless of what the roots of the quartic are. Proposed by Daniel Carter Answer: 16 Denote by Σ the sum of the products of one root raised to the a , a different root raised a,b,c,d to the b , a third root raised to the c , and the last root raised to the d . For example, if the 2 2 2 2 four roots are p, q, r, s , then Σ = p + q + r + s and Σ = pqrs . We have that 2 , 0 , 0 , 0 1 , 1 , 1 , 1 U = − P Σ , M = P Σ , A = − P Σ , and C = P Σ . Then one can see 1 , 0 , 0 , 0 1 , 1 , 0 , 0 1 , 1 , 1 , 0 1 , 1 , 1 , 1 2 2 2 M = P (Σ + 2Σ + 6Σ ) and U A = P (Σ + 4Σ ), so the expression 2 , 2 , 0 , 0 2 , 1 , 1 , 0 1 , 1 , 1 , 1 2 , 1 , 1 , 0 1 , 1 , 1 , 1 2 2 M − 2 U A + zP C is equal to P (Σ + ( z − 2)Σ ). 2 , 2 , 0 , 0 1 , 1 , 1 , 1 4 Taking p, q, r, s arbitrarily close to each other makes Σ close to 6 p and Σ close to 2 , 2 , 0 , 0 1 , 1 , 1 , 1 4 2 4 p , so this expression is arbitrarily close to P ( z + 4) p . Thus if z < − 4 this can be negative. Also, by AM-GM and the fact that the roots are all different we have Σ / 6 > Σ , so 2 , 2 , 0 , 0 1 , 1 , 1 , 1 2 if z ≥ − 4, the expression is positive. Thus z = − 4 and our answer is ( − 4) = 16. 2020 X 4 kπ