1a)Let be an odd positive integer. Let prove that .
1b)Let and an odd positive integer. Prove that .
2)Each man of men throws his wallet on the table, then every one picks up a wallet randomly. Find the probability the every person takes the wrong wallet.
3)Let and be circles with inside and tangent at a point on . Let ( ) be circles in between and and tangent to and and to eachother adjacent circle. Let be the points of tangency of these circles with their neighbors. Prove that all lie on a common circle.
(Note: The solution I know does not use elementary geometry).