\textbf{3}

Consider a coordinate plane with origin O. For two points $\mathrm{S}(x_1,\ y_1)$, $\mathrm{T}(x_2,\ y_2)$ on the coordinate plane, we define that point S is \textit{sufficiently far} from point T if
$$|x_1 - x_2| \geqq 1 \quad \text{or} \quad |y_1 - y_2| \geqq 1$$
holds.

Let $D$ be the square region defined by the inequalities $0 \leqq x \leqq 3$, $0 \leqq y \leqq 3$, and consider two of its vertices $\mathrm{A}(3,\ 0)$, $\mathrm{B}(3,\ 3)$. Furthermore, let P be a point satisfying both of the following conditions (i) and (ii).

\begin{itemize}
  \item[(i)] Point P is in region $D$, and lies on the parabola $y = x^2$.
  \item[(ii)] Point P is sufficiently far from each of the 3 points O, A, B.
\end{itemize}

Let $a$ be the $x$-coordinate of point P.

\begin{itemize}
  \item[(1)] Find the range of possible values of $a$.
  \item[(2)] Find the area $f(a)$ of the region where a point Q satisfying both of the following conditions (iii) and (iv) can exist.
    \begin{itemize}
      \item[(iii)] Point Q is in region $D$.
      \item[(iv)] Point Q is sufficiently far from each of the 4 points O, A, B, P.
    \end{itemize}
  \item[(3)] Suppose $a$ varies over the range found in (1). Find the value of $a$ that minimizes $f(a)$ from (2).
\end{itemize}



%% Page 4