I recently took the AIME, and the following question was one I was not able to answer:
On square $ABCD$, points $E,F,G,$ and $H$ lie on sides $\overline{AB}$,$\overline{BC}$,$\overline{CD}$, and $\overline{DA}$, respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34$. Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P$, and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411$. Find the area of square $ABCD$.
After drawing the shape and trying a few things, I drew a blank. I tried to set up a system of equations with the areas of the smaller quadrilaterals and their ratios, but that didn't work either. The solution is not up on the aops website yet, but I would really like to know it. Any help would be appreciated.