1. x and y are two natural numbers such that 3x² + x = 4y² +y . Prove that x - y is the square of a whole number.

2. Assume that KI and KN are the two tangents drawn from K onto the circle C . M is an arbitrary point on the extension of KN (near N) and P is the other extension point of the circle C with the circumcircle of KLM . Q is the foot of the altitude drawn from N onto ML . Prove that the measure of the angle MPQ is two times the angle KML .

3. Consider an n x n matrix of 0 , +1 and -1 , such that in each row and each column, there exists only one +1 and one -1 . Prove that by a finite number of changing columns with eachother and rows with eachother, we can change the places of +1's with -1's and vice versa.

4. x₁ , x₂, x₃ and x₄ are four positive real numbers such that  x₁x₂x₃x₄ = 1 . Prove that :

5. In the triangle ABC , B and C are acute angles. The altitude of the triangle drawn from Aintersects BC at D . The bisectors of the angles B and C intersect AD at E and F respectively . If BE = CF , Prove that the triangle ABC is isosceles.

6. Find the largest p such that a and b are two natural numbers and p = b/4 ((2a-b)/(2a+b))^1/2 is a prime number.