1. In nonisosceles triangle , let the excenters of the triangle opposite and be and respectively. Let the external angle bisector of intersect the circumcircle of triangle again at . Prove that |
2. Mr. Fat moves around on the lattice points according to the following rules: From point he may move to any of the points and . Show that if he starts at he can never get to .
3. Prove that for .
4. Let be an infinite sequence of real numbers. Prove that there exists an increasing sequence of positive integers such that the sequence is either nondecreasing or nonincreasing.
Time: 4 hours.
Each problem is worth 7 points.