Given that, $x_{0} = \text{max} (x_{1}, x_{2}, \dots, x_{12})$
That means,
- $x_{0} \geq x_{1}$
- $x_{0} \geq x_{2}$
- $x_{0} \geq x_{3}$
- $x_{0} \geq x_{4}$
- $x_{0} \geq x_{5}$
- $x_{0} \geq x_{6}$
- $x_{0} \geq x_{7}$
- $x_{0} \geq x_{8}$
- $x_{0} \geq x_{9}$
- $x_{0} \geq x_{10}$
- $x_{0} \geq x_{11}$
- $x_{0} \geq x_{12}$
Now, $x_{1} + x_{2} + \dots + x_{12} = 100$
$ \Rightarrow x_{0} + x_{0} + \dots + x_{0} \geq 100$
$ \Rightarrow 12 x_{0} \geq 100 $
$ \Rightarrow x_{0} \geq \frac{100}{12} $
$ \Rightarrow x_{0} = \left \lceil \frac{100}{12} \right \rceil $
$ \Rightarrow x_{0} = \left \lceil 8.33 \right \rceil$
$ \Rightarrow \boxed{x_{0} = 9}$
Correct Answer$: 9$