Try the followings:
The eleven members of a cricket team are numbered from . In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two?
Compute the maximum area of a rectangle which can be inscribed in a triangle of area .
Let where . Suppose . Prove that for all .
Let be the sequence of all the positive integers which do not contain the digit . Write for this sequence. By comparing with a geometric series, show that .
There are people in a queue waiting to enter a hall. The hall has exactly seats numbered from to . The first person in the queue enters the hall, chooses any seat and sits there. The -th person in the queue where can be , enters the hall after -th person is seated. He sits in seat number if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which seats can be filled up, provided the -th person occupies seat number .
Let . Consider all subsets of consisting of elements. Let denote the arithmetic mean of the smallest elements of these subsets. Prove that .