Problem

Source: RMO KV 2024 Q2

Tags: algebra



Show that there do not exist non-zero real numbers $a,b,c$ such that the following statements hold simultaneously: $\bullet$ the equation $ax^2 + bx + c = 0$ has two distinct roots $x_1,x_2$; $\bullet$ the equation $bx^2 + cx + a = 0$ has two distinct roots $x_2,x_3$; $\bullet$ the equation $cx^2 + ax + b = 0$ has two distinct roots $x_3,x_1$. (Note that $x_1,x_2,x_3$ may be real or complex numbers.)