ISI Entrance Paper BMath 2005 – from Indian Statistical Institute’s Entrance

Also see: ISI and CMI Entrance Course at Cheenta

- For any , prove that:-

Also compute integral part of . - Let and for all . Define : Compute
- Let ABCD be a quadrilateral such that the sum of a pair of opposite sides equals the sum of other pair of opposite sides (AB+CD=AD+BC). Prove that the circles inscribed in triangles ABC and ACD are tangent to each other.
- For a set S we denote its cardinality by |S|. Let be non-negative integers. Let (respectively ) be the set of all k-tuples of integers such that for all i and is even (respectively odd). Show that .
- Find the point in the closed unit disc at which the function f(x,y)=x+y attains its maximum .
- Let Show that , for , the polynomial p(t) has exactly one root in the interval
- Let M be a point in the triangle ABC such that

Show that the locus of all such points is a straight line. - In how many ways can one fill an n*n matrix with +1 and -1 so that the product of the entries in each row and each column equals -1?

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