IMO预选题1991.pdf

Problems Shortlisted to the 1991 IMO Jury 1. Let be a triangle and be an interior point. Let the feet of the perpendiculars from to , be , respectively, and let the feet of the perpendiculars from 1 2 to , be , respectively. Show that = . 1 2 1 2 2 1 2. Let be an acute-angled triangle. Let be the midpoint of and be the point on such that = . Let be the foot of the perpendicular from to . The lines through perpendicular to and meet and at and respectively. Show that is tangent to the circle through at . 3. If is a point on the circumcircle of the triangle , show that the feet of the perpendiculars from to the lines , and are collinear. Denote the line by . If is an inscribed hexagon, show that the lines , , , are concurrent if and only if is a rectangle. ˆ 4. Let be a triangle with = 60 and incenter . Let be a point on BC so that 3 = . The point is chosen on so that . Show that = . 5. Let be the circumcenter of the tetrahedron . Let be the midpoints of , , respectively. If + = + , + = + and + = + the show that = = . 6. Given a set of points in the plane, no three collinear, show that we can find a set of 2 5 points such that a point of lies in the interior of every triangle whose vertices belong to . 7. A graph has 1991 points and every point has degree at least 1593. Show that there is a complete subgraph with 6 vertices, i.e. there are six points, each of which is joined to the others. Is 1593 the smallest degree for w