Question 116
Quadratic polynomials $f_1(x), \dots, f_{100}(x)$ have equal leading coefficients, equal coefficients of $x$, but distinct constant terms; each of these polynomials has two real roots. Let $x_i$ be some root of $f_i(x)$. Determine all possible values of the sum $f_2(x_1) + f_3(x_2) + \dots + f_{100}(x_{99}) + f_1(x_{100})$. (N. Agakhanov)