Question 68
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 $x = 1$.
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 $x = 1$.