**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 .