roots of a cubic
Source: Hungary-Israel Binational Olympiad 2009, Problem 2
August 17, 2009
algebra unsolvedalgebra
Problem Statement
Denote the three real roots of the cubic x^3 \minus{} 3x \minus{} 1 \equal{} 0 by , , in order of increasing magnitude. (You may assume that the equation in fact has three distinct real roots.) Prove that x_3^2 \minus{} x_2^2 \equal{} x_3 \minus{} x_1.