Problem 1. Let be a triangle. The incircle of triangle touches side at Let segment meet the incircle again at Segments meet the incircle at respectively. Show that lines are concurrent.
Problem 2. Given acute triangle with , let be the midpoint of . is a point in triangle such that . Let be the circumcenters of respectively. Prove that line passes through the midpoint of .
Problem 3. Let be a cyclic kite (i.e. is a perpendicular chord onto the diameter ) and the midpoint of . The perpendicular from onto intersects at . Prove that is tangent to the circle .
Problem 4. In triangle , let be the incenter and let be the excenter opposite . Suppose that meets and the circumcircle of triangle at and , respectively. Let be the midpoint of arc of the circumcircle of triangle . Let lines and intersect the circumcircle of triangle again at and , respectively. Prove that , , and are collinear.
Problem 5. Point lies on diagonal of parallelogram . Line intersects side and line at points and , respectively. Let be the circle with center and radius and be the circumcircle of triangle . intersect at and . Prove that and are tangent to .
Problem 6. In an acute-angled triangle , and are the altitudes and the orthocentre. Lines and meet at . The line passing through and parallel to meets lines and at and , respectively. Prove that the circumcircle of the triangle bisects the side .
Problem 7. Let an acute triangle so that and let the orthocenter of . Points and are in the sides respectively so that are collinear and . Prove that if is the midpoint of , then is perpendicular to the straight line that joins with the second intersections (different of ) of the circumcircles of the triangles and .