1. Let be an infinite sequence of positive integers. Prove that there exists a unique integer such that
2. Let be an integer. Consider an chessboard consisting of unit squares. A configuration of rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer such that, for each peaceful configuration of rooks, there is a square which does not contain a rook on any of its unit squares.
3. Convex quadrilateral has . Point is the foot of the perpendicular from to . Points and lie on sides and , respectively, such that lies inside triangle and
Prove that line is tangent to the circumcircle of triangle .