<s>[INST] 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.
 [/INST] Adding $1$ to each equation, we get that 
$$\left \{\begin{matrix}a + b + ab +1 = 16 \\ 
b + c + bc+1 = 100 \\
c + a + ca+1 = 400\end{matrix}\right. $$
Factoring, we get
$$\left \{\begin{matrix}(b+1)(a+1) = 16 \\ 
(b+1)(c+1) = 100 \\
(c+1)(a+1) = 400\end{matrix}\right. $$
Dividing equation 2 from equation 3 and multiplying the result with equation 1, we get that $(a+1)^2=64$, meaning $a+1=8$. Equation one gives us $b+1=2$, and equation three gets us $c+1=50$, getting that
$$a+b+c+bc=7+1+49+49=106$$
</s>