Question 189
Let $a_1, a_2, a_3, \dots, a_{10}$ and $b_1, b_2, \dots, b_{10}$ be real numbers such that the roots of these 10 polynomials
$$
x^2 + a_1x + b_1,\ x^2 + a_2x + b_2,\ \dots,\ x^2 + a_{10}x + b_{10}
$$
are all the integers $\pm 1, \pm 2, \dots, \pm 10$ (in some order).
a) What is the maximum number of odd values among $a_1, b_1, \dots, a_{10}, b_{10}$?
b) Find the minimum and maximum values of the sum $b_1 + b_2 + \dots + b_{10}$.
$$
x^2 + a_1x + b_1,\ x^2 + a_2x + b_2,\ \dots,\ x^2 + a_{10}x + b_{10}
$$
are all the integers $\pm 1, \pm 2, \dots, \pm 10$ (in some order).
a) What is the maximum number of odd values among $a_1, b_1, \dots, a_{10}, b_{10}$?
b) Find the minimum and maximum values of the sum $b_1 + b_2 + \dots + b_{10}$.