Here's a problem I found in the Problems section of a
math journal. The way I read it, it is not true.

Let ABCD be a convex quadrilateral. Prove that

there exists a point P inside ABCD such that
[PAB]=[PBC]=[PCD]=[PDA],

    if and only if

the diagonals bisect each other.

Here [XYZ] denotes the area of triangle XYZ.