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 
.
, 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,
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 .
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 alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of 
by 
. And at 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.
. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of by . And at 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 .
, 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.
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.
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