\noindent\textbf{1}

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Let $a, b, c, p$ be real numbers. Suppose that the set of real numbers $x$ satisfying all of the inequalities
$$ax^2 + bx + c > 0, \quad bx^2 + cx + a > 0, \quad cx^2 + ax + b > 0$$
coincides with the set of real numbers $x$ satisfying $x > p$.

\begin{itemize}
  \item[(1)] Show that $a, b, c$ are all non-negative.
  \item[(2)] Show that at least one of $a, b, c$ is $0$.
  \item[(3)] Show that $p = 0$.
\end{itemize}



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When points P, Q, R in a plane are not collinear, we denote the area of the triangle with these three vertices by $\triangle\mathrm{PQR}$. When P, Q, R are collinear, we set $\triangle\mathrm{PQR} = 0$.

Let A, B, C be three points in a plane with $\triangle\mathrm{ABC} = 1$. A point X in this plane satisfies
$$2 \leqq \triangle\mathrm{ABX} + \triangle\mathrm{BCX} + \triangle\mathrm{CAX} \leqq 3.$$

Find the area of the region in which X can move.



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