ISI Entrance Paper BMath 2007 – from Indian Statistical Institute’s Entrance

Also see: ISI and CMI Entrance Course at Cheenta

- 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 has a non-zero solution in rational numbers . Prove that for any rational number t , there is a solution of the equation .
- For a natural number n>1 , consider the n-1 points on the unit circle . 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 and are two integers such that and , then prove that divides 2.(b) Prove that the number of distinct integer roots of 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 . Prove that .

Solution - Let 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 ?

please send me the solution of the forth no question.

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