1. Let n be a positive integer . If n has odd number of divisors ( other than 1 and n ) , then show that n is a perfect square .

 

    1. Let a and b be two non-zero rational numbers such that the equation \mathbf{ax^2+by^2=0} has a non-zero solution in rational numbers . Prove that for any rational number t , there is a solution of the equation \mathbf{ax^2+by^2=t}.

 

    1. For a natural number n>1 , consider the n-1 points on the unit circle \mathbf{e^{\frac{2\pi ik}{n}} (k=1,2,...,n-1)} . Show that the product of the distances of these points from 1 is n.

 

    1. Let ABC be an isosceles triangle with AB=AC=20 . Let P be a point inside the triangle ABC such that the sum of the distances of P to AB and AC is 1 . Describe the locus of all such points inside triangle ABC.

 

    1. Let P(X) be a polynomial with integer coefficients of degree d>0.(a) If \mathbf{\alpha} and \mathbf{\beta} are two integers such that \mathbf{P(\alpha)=1} and \mathbf{P(\beta)=-1} , then prove that \mathbf{|\beta - \alpha|} divides 2.

      (b) Prove that the number of distinct integer roots of \mathbf{P^2(x)-1} is at most d+2.

 

    1. In ISI club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does ISI club have?

 

    1. Let \mathbf{0\leq \theta\leq \frac{\pi}{2}} . Prove that mathbf{sin \theta \geq \frac{2\theta}{\pi}}.
      Solution

 

    1. Let \mathbf{P:\mathbb{R} to \mathbb{R}} be a continuous function such that P(X)=X has no real solution. Prove that P(P(X))=X has no real solution.
      Solution

 

    1. In a group of five people any two are either friends or enemies , no three of them are friends of each other and no three of them are enemies of each other . Prove that every person in this group has exactly two friends .

 

  1. The eleven members of a cricket team are numbered 1,2,…,11. 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 ?