Let $u$ and $v$ be [integers](https://artofproblemsolving.com/wiki/index.php/Integer) satisfying $0 < v < u$. Let $A = (u,v)$, let $B$ be the [reflection](https://artofproblemsolving.com/wiki/index.php/Reflection) of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of [pentagon](https://artofproblemsolving.com/wiki/index.php/Pentagon) $ABCDE$ is $451$. Find $u + v$.
Solution 1

Since $A = (u,v)$, we can find the coordinates of the other points: $B = (v,u)$, $C = (-v,u)$, $D = (-v,-u)$, $E = (v,-u)$. If we graph those points, we notice that since the latter four points are all reflected across the x/y-axis, they form a rectangle, and $ABE$ is a triangle. The area of $BCDE$ is $(2u)(2v) = 4uv$ and the area of $ABE$ is $\frac{1}{2}(2u)(u-v) = u^2 - uv$. Adding these together, we get $u^2 + 3uv = u(u+3v) = 451 = 11 \cdot 41$. Since $u,v$ are positive, $u+3v>u$, and by matching factors we get either $(u,v) = (1,150)$ or $(11,10)$. Since $v < u$ the latter case is the answer, and $u+v = \boxed{021}$.

Solution 2
We find the coordinates like in the solution above: $A = (u,v)$, $B = (v,u)$, $C = (-v,u)$, $D = (-v,-u)$, $E = (v,-u)$. Then we apply the [Shoelace Theorem](https://artofproblemsolving.com/wiki/index.php/Shoelace_Theorem). \[A =  \frac{1}{2}[(u^2 + vu + vu + vu + v^2) - (v^2 - uv - uv - uv -u^2)] = 451\] \[\frac{1}{2}(2u^2 + 6uv) = 451\] \[u(u + 3v) = 451\]
This means that $(u,v) = (11, 10)$ or $(1,150)$, but since $v < u$, then the answer is $\boxed{021}$