Question 189
Let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be real numbers such that
$$
(a_1^2 + a_2^2 + \cdots + a_n^2 - 1)(b_1^2 + b_2^2 + \cdots + b_n^2 - 1)
\gt (a_1b_1 + a_2b_2 + \cdots + a_nb_n - 1)^2.
$$
Show that $a_1^2 + a_2^2 + \cdots + a_n^2 \gt 1$ and $b_1^2 + b_2^2 + \cdots + b_n^2 \gt 1$.
$$
(a_1^2 + a_2^2 + \cdots + a_n^2 - 1)(b_1^2 + b_2^2 + \cdots + b_n^2 - 1)
\gt (a_1b_1 + a_2b_2 + \cdots + a_nb_n - 1)^2.
$$
Show that $a_1^2 + a_2^2 + \cdots + a_n^2 \gt 1$ and $b_1^2 + b_2^2 + \cdots + b_n^2 \gt 1$.