Question 61
Given $10$ quadratic equations
$$
x^{2} + a_{1} x + b_{1} = 0,\quad x^{2} + a_{2} x + b_{2} = 0,\quad \ldots,\quad x^{2} + a_{10} x + b_{10} = 0
$$
each of them has two distinct real roots and the set of all roots is $S = \{ \pm 1, \pm 2, \ldots, \pm 10 \}$. Find the minimum value of the sum $T = b_{1} + b_{2} + \cdots + b_{10}$.
$$
x^{2} + a_{1} x + b_{1} = 0,\quad x^{2} + a_{2} x + b_{2} = 0,\quad \ldots,\quad x^{2} + a_{10} x + b_{10} = 0
$$
each of them has two distinct real roots and the set of all roots is $S = \{ \pm 1, \pm 2, \ldots, \pm 10 \}$. Find the minimum value of the sum $T = b_{1} + b_{2} + \cdots + b_{10}$.