Answer: 13
Note that such $E, F$ exist if and only if
$$
\frac{[A D C]}{[A D B]}=2 .
$$
([] denotes area.) Since $A D$ is the angle bisector, and the ratio of areas of triangles with equal height is the ratio of their bases,
$$
\frac{A C}{A B}=\frac{D C}{D B}=\frac{[A D C]}{[A D B]} .
$$

Hence (1) is equivalent to $A C=2 A B=14$. Then $B C$ can be any length $d$ such that the triangle inequalities are satisfied:
$$
\begin{aligned}
d+7 & >14 \\
7+14 & >d
\end{aligned}
$$

Hence $7<d<21$ and there are 13 possible integral values for $B C$.
Algebra \& Geometry Individual Test
2. Let $A B C$ be a triangle such that $A B=7$, and let the angle bisector of $\angle B A C$ intersect line $B C$ at $D$. If there exist points $E$ and $F$ on sides $A C$ and $B C$, respectively, such that lines $A D$ and $E F$ are parallel and divide triangle $A B C$ into three parts of equal area, determine the number of possible integral values for $B C$.