Find a + b + c + bc. My math trainer friend ask my help to look for the shortest possible solution for this problem:
Let $a$, $b$ and $c$ be positive integers such that 
$$\left \{\begin{matrix}a + b + ab = 15 \\ 
b + c + bc = 99 \\
c + a + ca = 399\end{matrix}\right. $$

Find $a + b + c + bc.$

I tried elimination but it took us for about 3 min to do it. This question was intended for 15 seconds only.

Also elimination leads to the result (as the OP wanted), namely we obtain exactly two solutions over any field of characteristic zero:
$$
(a,b,c)=(-9,-3,-51),(7,1,49).
$$
Then $a+b+c+bc=7+1+49=106$, since only the positive solution was asked.
