Problem 4.   Let Γ be a circle with centre I, and ABCD a convex quadrilateral such that each of the segments AB, BC, CD and DA is tangent to Γ. Let Ω be the circumcircle of the triangle AIC. The extension of BA beyond A meets Ω at X, and the extension of BC beyond C meets Ω at Z. The extensions of AD and CD beyond D meet Ω at Y  and T , respectively. Prove that

AD + DT + T X + XA = CD + DY + Y Z + ZC.

proposed by Dominik Burek and Tomasz Ciesla, Poland