- 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 .

- 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 \(\mathbf{ax^2+by^2=t}\).

- 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.

- 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.

- 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.

- 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?

- Let \(\mathbf{0\leq \theta\leq \frac{\pi}{2}}\) . Prove that \(mathbf{sin \theta \geq \frac{2\theta}{\pi}}\).

Solution

- 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

- 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 .

- 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 ?

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