2021 AMC 12D Question 7 - thichhoctoan.com

Question 7

Which of the following conditions is sufficient to guarantee that integers $x,$ $y,$ and $z$ satisfy the equation $$
\begin{aligned} &x(x - y) + y(y - z) \\ &\quad {}+ z(z - x) = 1? \end{aligned}
$$

$x \gt y$ and $y = z$
$x = y - 1$ and $y = z - 1$
$x = z + 1$ and $y = x + 1$
$x = z$ and $y - 1 = x$
$x + y + z = 1$