1. Two circles and of different radii intersect at two points and , let and be two points on and , respectively, such that is the midpoint of the segment . The extension of meets at another point , the extension of meets at another point . Let and be the perpendicular bisectors of and , respectively.
Show that and have a unique common point (denoted by ).
Prove that the lengths of , and are the side lengths of a right triangle.
2. Find all nonempty sets of integers such that for all (not necessarily distinct) .
3. Find all positive real numbers with the following property: there exists an infinite set of real numbers such that the inequality
holds for all (not necessarily distinct) , all real numbers and all positive real numbers .
4. , there are finite sets satisfy that for any , . Try to find the the minimum of .
5. For any positive integer and , denote , where , define . are positive integers, and for any . Prove that if , then for any positive integer .
6. are positive integers, find the minimum positive integer satisties that if is a set of integers contains a complete residue system of and , then there exit a nonempty set , that .