Problem

Source: 2018 JBMO TST - Turkey, P1

Tags: algebra



Let $a, b, c$ be distinct real numbers and $x$ be a real number. Given that three numbers among $ax^2+bx+c, ax^2+cx+b, bx^2+cx+a, bx^2+ax+c, cx^2+ax+b, cx^2+bx+a$ coincide, prove that $x=1$.