AoPS Wiki:Problem of the Day/June 11, 2011

Consider the unit square $ABCD$. Let points $E,F,G,$ and $H$ be the midpoints of sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{AD},$ respectfully. Then, segments $\overline{EG}$ and $\overline{FH}$ intersect at point $X$. Points $J,K,L,$ and $M$ are the midpoints of $\overline{XE},\overline{XF},\overline{XG},$ and $\overline{XH},$ also respectfully. Draw triangles $\triangle AJM$, $\triangle BKJ$, $\triangle CLK$, and $\triangle DML$. Points $A$, $B$, $C$, and $D$ fold up to form the apex of a square pyramid with base $JKLM$. Find the area of the square pyramid.