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.
Note: alternatively, we can just ask “Show that there exists a unique persistent number, and determine its value”.