RU-SPB 2012 Problem 101

$\begin{cases} x^3-ax^2+b^3=0 x^3-bx^2+c^3=0 x^3-cx^2+a^3=0 \end{cases}$ Prove that system hasn`t solutions if $a,b,c$ are different.

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.