Select Page

 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.