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2022 Alg/NT Div 2 P7

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February 28, 2022
algebranumber theory

Problem Statement

For polynomials P(x)=anxn+⋯+a0P(x) = a_nx^n + \cdots + a_0, let f(P)=an⋯a0f(P) = a_n\cdots a_0 be the product of the coefficients of PP. The polynomials P1,P2,P3,QP_1,P_2,P_3,Q satisfy P1(x)=(x−a)(x−b)P_1(x) = (x-a)(x-b), P2(x)=(x−a)(x−c)P_2(x) = (x-a)(x-c), P3(x)=(x−b)(x−c)P_3(x) = (x-b)(x-c), Q(x)=(x−a)(x−b)(x−c)Q(x) = (x-a)(x-b)(x-c) for some complex numbers a,b,ca,b,c. Given f(Q)=8f(Q) = 8, f(P1)+f(P2)+f(P3)=10f(P_1) + f(P_2) + f(P_3) = 10, and abc>0abc>0, find the value of f(P1)f(P2)f(P3)f(P_1)f(P_2)f(P_3).
Proposed by Justin Hsieh
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