Question 76
Call a number $T$ persistent if the following holds: Whenever $a, b, c, d$ are real numbers different from $0$ and $1$ such that
$$
a + b + c + d = T
$$
and
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = T
$$
we also have
$$
\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} + \frac{1}{1-d} = T.
$$
a. If $T$ is persistent, prove that $T$ must be equal to $2$.
b. Prove that $2$ is persistent.
$$
a + b + c + d = T
$$
and
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = T
$$
we also have
$$
\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} + \frac{1}{1-d} = T.
$$
a. If $T$ is persistent, prove that $T$ must be equal to $2$.
b. Prove that $2$ is persistent.