Problem 1: Define a sequence 
as: 
Prove that this sequence has a finite limit as 
Also determine the limit.
and 
be two sequences of numbers, and let 
be an integer greater than 
Define 
Prove that if the quadratic expressions 
do not have any real roots, then all the remaining polynomials also don’t have real roots.
be a cyclic quadrilateral with circumcentre 
and the pair of opposite sides not parallel with each other. Let 
and 
Denote, by 
the intersection of the angle bisectors of 
and 
and 
and 
Suppose that the four points 
are distinct. are concyclic. Find the centre of this circle, and denote it as 
Prove that 
are collinear.
be a natural number. There are boys and girls standing in a line, in any arbitrary order. A student 
will be eligible for receiving candies, if we can choose two students of opposite sex with standing on either side of in ways. Show that the total number of candies does not exceed 
and 
boys. There are 
chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met:
and 
there are at least three boys.
and 
there are at least one boy and most four boys.
and 
where is a divisor of 
and is a divisor of 
Prove that and are the terms of the series of natural numbers 
defined by
such that: there exist real number :
then 
