ILL 1971 Problem 31

Determine whether there exist distinct real numbers $a, b, c, t$ for which:

(i) the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$

(ii) the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$

(iii) the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$