1. Let A,B be two distinct points on a given circle O and let M be the midpoint of arc AB,C be an arbitrary point outside the circle O.CS and CT are two lines passing through C and tangent to the circle O at S and T respectively.
Let , two lines passing through E and F respectively and perpendicular to AB intersect OS and OT at X and Y respectively. A line through C cuts the circle O at P and Q ,let , Z be the center of the circumcircle of .Show that X,Y,Z are collinear.
2. A national number is called “good” if it satisfies: with being positive integers, and there exists constant numbers such that for any integer ,
Find all the “good” numbers.
3.There are points arbitrarily on the circle with its diameter being . Let denote the number of triangles whose vertices are three of the points and the length of its sides is no less than . Fine the maximum of .
4. Fine all functions such that
.Here we denote by be the positive rational number set.
5. Let be real numbers satisfing:
and . Prove that for any vectors in the plane, there exists a permutations of the numbers such that
6.Let be a positive integer, let be a subset of ,satisfing for any two numbers ,the least common multiple of not more than .
Show that .