Problem (Geometry):
Points $A, R, M$, and $L$ are consecutively the midpoints of the sides of a square whose area is 650. The coordinates of point $A$ are $(11,5)$. If points $R, M$, and $L$ are all lattice points, and $R$ is in Quadrant I, compute the number of possible ordered pairs $(x, y)$ of coordinates for point $R$.

Solution:
Write $x=11+c$ and $y=5+d$. Then $A R^{2}=c^{2}+d^{2}=\frac{1}{2} \cdot 650=325$. Note that $325=18^{2}+1^{2}=17^{2}+6^{2}=15^{2}+10^{2}$. Temporarily restricting ourselves to the case where $c$ and $d$ are both positive, there are three classes of solutions: $\{c, d\}=\{18,1\},\{c, d\}=\{17,6\}$, or $\{c, d\}=\{15,10\}$. In fact, $c$ and $d$ can be negative, so long as those values do not cause $x$ or $y$ to be negative. So there are 10 solutions:

| $(c, d)$ | $(x, y)$ |
| :---: | :---: |
| $(18,1)$ | $(29,6)$ |
| $(18,-1)$ | $(29,4)$ |
| $(1,18)$ | $(12,23)$ |
| $(-1,18)$ | $(10,23)$ |
| $(17,6)$ | $(28,11)$ |
| $(6,17)$ | $(17,22)$ |
| $(-6,17)$ | $(5,22)$ |
| $(15,10)$ | $(26,15)$ |
| $(10,15)$ | $(21,20)$ |
| $(-10,15)$ | $(1,20)$ |

Final answer: 10