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System of equations with a parameter

Source: 2007 Bulgarian Autumn Math Competition, Problem 8.1

March 17, 2022

algebrasystem of equationsparameterparameterization

Problem Statement

Determine all real

a

, such that the solutions to the system of equations

\begin{cases} \frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\ (2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a \end{cases}

form an interval with length

\frac{32}{225}