Question 20
Let $x, y, z$ be real numbers satisfying
$$
\frac{1}{x} + y + z = x + \frac{1}{y} + z = x + y + \frac{1}{z} = 3.
$$
The sum of all possible values of $x + y + z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n) = 1$. Find $100m + n$.
$$
\frac{1}{x} + y + z = x + \frac{1}{y} + z = x + y + \frac{1}{z} = 3.
$$
The sum of all possible values of $x + y + z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n) = 1$. Find $100m + n$.