BG-MO 1984 Problem 2

The diagonals of a trapezoid $ABCD$ with bases $AB$ and $CD$ intersect in a point $O$ , and $AB/CD=k>1$ . The bisectors of the angles $AOB,BOC,COD,DOA$ intersect $AB,BC,CD,DA$ respectively at $K,L,M,N$ . The lines $KL$ and $MN$ meet at $P$ , and the lines $KN$ and $LM$ meet at $Q$ . If the areas of $ABCD$ and $OPQ$ are equal, find the value of $k$ .