Question 78
For what real numbers $\lambda$ are there positive numbers $a$, $b$, and $c$ such that
$$
a + \frac{1}{a} = 2(1 + \lambda),
$$
$$
c + \frac{1}{c} = 2(1 + \lambda),
$$
$$
b + \frac{1}{a} = 2(1 - \lambda),
$$
$$
c + \frac{1}{b} = 2(1 - \lambda)?
$$
For those values of $\lambda$, determine the corresponding values of $a$, $b$ and $c$.
$$
a + \frac{1}{a} = 2(1 + \lambda),
$$
$$
c + \frac{1}{c} = 2(1 + \lambda),
$$
$$
b + \frac{1}{a} = 2(1 - \lambda),
$$
$$
c + \frac{1}{b} = 2(1 - \lambda)?
$$
For those values of $\lambda$, determine the corresponding values of $a$, $b$ and $c$.