Question 161
Let $a, b, c, d$ be real numbers so that $c, d$ are not both $0$. Define the function
$$
m(x) = \frac{a x + b}{c x + d}
$$
on all real numbers $x$ except possibly $-d / c$, in the event that $c \neq 0$. Suppose that the equation $x = m(m(x))$ has at least one solution that is not a solution of $x = m(x)$. Find all possible values of $a + d$. Prove that your answer is correct.
$$
m(x) = \frac{a x + b}{c x + d}
$$
on all real numbers $x$ except possibly $-d / c$, in the event that $c \neq 0$. Suppose that the equation $x = m(m(x))$ has at least one solution that is not a solution of $x = m(x)$. Find all possible values of $a + d$. Prove that your answer is correct.