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

Also see: ISI and CMI Entrance Course at Cheenta

  1. 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).
  2. 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.
  3. Find all roots of the equation :-\mathbf{ 1-\frac{x}{1}+\frac{x(x-1)}{2!} - \cdots +(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0.}
  4. Let \mathbf{f:\mathbb{R} to \mathbb{R}} be a function that is a function that is differentiable n+1 times for some positive integer n . The \mathbf{i^{th}} derivative of f is denoted by \mathbf{f^{(i)}} . Suppose-\mathbf{f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0.}
    Prove that \mathbf{f^{(n+1)}(x)=0} for some \mathbf{x \in (0,1)}
  5. 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?
  6. 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).
  7. In a triangle ABC , D is a point on BC such that AD is the internal bisector of \mathbf{\angle A} . Now Suppose \mathbf{\angle B=2\angle C} and CD=AB . Prove that \mathbf{\angle A=72^0}
  8. Let S be the set of all integers k, \mathbf{1\leq k\leq n}, such that \mathbf{gcd(k,n)=1}. What is the arithmetic mean of the integers in S?