MathDB

Cubic Equation

Source: 2002 National High School Mathematics League, Exam Two, Problem 2

March 16, 2020

Problem Statement

For real numbers

a,b,c

and positive number

\lambda

such that three real roots

x_1,x_2,x_3

f(x)=x^3+ax^2+bx+c

satisfying:

(1) x_2-x_1=\lambda

;

(2) x_3>\frac{1}{2}(x_1+x_2)

. Find the maximum value of

\frac{2a^3+27c-9ab}{\lambda^3}