INMO 2004 Problem 1

Consider a convex quadrilateral $A B C D$, in which $K, L, M, N$ are the midpoints of the sides $A B$, $B C, C D, D A$ respectively. Suppose

(a) $B D$ bisects $K M$ at $Q$;

(b) $Q A=Q B=Q C=Q D$; and

(c) $L K / L M=C D / C B$.

Prove that $A B C D$ is a square.