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System of Quadratic Polynomials: sum equal to zero or not all are distinct

Source: Indonesian MO (INAMO) 2020, Day 1, Problem 2

October 13, 2020
quadraticsalgebrapolynomial2020Indonesia MOIndonesian MO

Problem Statement

Problem 2. Let P(x)=ax2+bx+cP(x) = ax^2 + bx + c where a,b,ca, b, c are real numbers. If P(a)=bc,P(b)=ac,P(c)=abP(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab then prove that (a−b)(b−c)(c−a)(a+b+c)=0.(a - b)(b - c)(c - a)(a + b + c) = 0.
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