AMC 12 A

Answer (B): The given condition means that either $x$ and $x^2$ represent the same angle or they represent supplementary angles. In the first case $x^2 = x + 360k$ for some integer $k$. By the quadratic formula,
$$x = \frac{1 \pm \sqrt{1 + 1440k}}{2}.$$
The least value of $x > 1$ occurs when $k = 1$ and the plus sign is used, in which case
$$x = \frac{1 + \sqrt{1441}}{2}.$$
In the second case $x^2 + x = 180(2k + 1)$ for some integer $k$. By the quadratic formula,
$$x = \frac{-1 \pm \sqrt{1 - 720(2k + 1)}}{2}.$$
The least value of $x > 1$ occurs when $k = -1$ and the plus sign is used, in which case
$$x = \frac{-1 + \sqrt{721}}{2}.$$
Because $27^2 = 729 > 721$ and $25^2 = 625 < 721$, the value of $x$ rounds up to $\frac{-1+27}{2} = 13$, which is smaller than the value of $x$ obtained from the first case.