Problem 1: Define a sequence
as: 
Prove that this sequence has a finite limit as
Also determine the limit.
Problem 2: Let

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.
Problem 3: Let

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.
(a) Show that the four points

are concyclic. Find the centre of this circle, and denote it as

(b) Let

Prove that

are collinear.
Problem 4: Let

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

Problem 5: For a group of 5 girls, denoted as

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:
(a) Each chair has a proper seat.
(b) The order, from left to right, of the girls seating is

(c) Between

and

there are at least three boys.
(d) Between

and

there are at least one boy and most four boys.
How many such arrangements are possible?
Problem 6: Consider two odd natural numbers

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

Problem 7: Find all

such that:
(a) For every real number

there exist real number

:

(b) If

then

(c)
