Question 130
Determine if there exists non-zero real numbers $a_1, a_2, \dots, a_{10}$ such that
$$ \left(a_1 + \frac{1}{a_1}\right) \cdots \left(a_{10} + \frac{1}{a_{10}}\right) = \left(a_1 - \frac{1}{a_1}\right) \cdots \left(a_{10} - \frac{1}{a_{10}}\right). $$
Do such non-zero real numbers exist? $a_1, a_2, \dots, a_{10}$, what
$$
\left(a_1 + \frac{1}{a_1}\right) \cdots \left(a_{10} + \frac{1}{a_{10}}\right) = \left(a_1 - \frac{1}{a_1}\right) \cdots \left(a_{10} - \frac{1}{a_{10}}\right)?
$$
$$ \left(a_1 + \frac{1}{a_1}\right) \cdots \left(a_{10} + \frac{1}{a_{10}}\right) = \left(a_1 - \frac{1}{a_1}\right) \cdots \left(a_{10} - \frac{1}{a_{10}}\right). $$
Do such non-zero real numbers exist? $a_1, a_2, \dots, a_{10}$, what
$$
\left(a_1 + \frac{1}{a_1}\right) \cdots \left(a_{10} + \frac{1}{a_{10}}\right) = \left(a_1 - \frac{1}{a_1}\right) \cdots \left(a_{10} - \frac{1}{a_{10}}\right)?
$$