<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] $(a+1)(b+1) = 16 $
$(b+1)(c+1) = 100$
$(c+1)(a+1) = 400$
Multiply all $3$ equations to get value of $(a+1)(b+1)(c+1) = 800$ then
$c+1 = 50,\quad a+1 = 8,\quad b+1 = 2$ , required is $a+(99) = 106$
</s>