1 In a triangle  , let  be the point on the segment  such that  . Suppose that the points  ,  and the centroids of triangles  and  lie on a circle. Prove that  . 2 Let  be a natural number. Prove that,  is even. 3 Let  be natural numbers with  . Suppose that the sum of their greatest common divisor and least common multiple is divisble by  . Prove that the quotient is at most  . When is this quotient exactly equal to  4 Written on a blackboard is the polynomial  . Calvin and hobbes take turns alternatively(starting with Calvin) in the following game. During his turns alternatively(starting with Calvin) in the following game. During his turn, Calvin should either increase or decrese the coeffecient of  by  . And during this turn, Hobbes should either increase or decrease the constant coefficient by  . Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy. 5 In a acute-angled triangle  , a point  lies on the segment  . Let  denote the circumcentres of triangles  and  respectively. Prove that the line joining the circumcentre of triangle  and the orthocentre of triangle  is parallel to  . 6 Let  be a natural number. Let  , and define  to be the set of all those elements of  which belong to exactly one of  and  . Show that  , where  is a collection of subsets of  such that for any two distinct elements of  of  we have  . Also find all such collections  for which the maximum is attained.