ISI Entrance Paper BMath 2006 – from Indian Statistical Institute’s B.Stat Entrance

Also see: ISI and CMI Entrance Course at Cheenta

- Bishops on a chessboard move along the diagonals ( that is, on lines parallel to the two main diagonals). Prove that the maximum number of non-attacking bishops on an n*n chessboard is 2n-2. (Two bishops are said to be attacking if they are on a common diagonal).
- Prove that there is no non-constant polynomial P(x) with integer coefficients such that P(n) is a prime number for all positive integers n.
- Find all roots of the equation :-
- Let be a function that is a function that is differentiable n+1 times for some positive integer n . The derivative of f is denoted by . Suppose-

Prove that for some - A domino is a 2 by 1 rectangle. For what integers m and n can we cover an m*n rectangle with non-overlapping dominoes?
- You are standing at the edge of a river which is 1 km wide. You have to go to your camp on the opposite bank. The distance to the camp from the point on the opposite bank directly across you is 1 km. You can swim at 2 km/hr and walk at 3 km-hr. What is the shortest time you will take to reach your camp? (Ignore the speed of the river and assume that the river banks are straight and parallel).
- In a triangle ABC , D is a point on BC such that AD is the internal bisector of . Now Suppose and CD=AB . Prove that
- Let S be the set of all integers k, , such that . What is the arithmetic mean of the integers in S?

Problems 1 and 3 requested