1997 Mathematical Olympiad of Iran

First Day
April 24th, 1997
Time : 4 hours

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.

 Second Day
April 25th, 1997
Time : 4 hours

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

x13 + x23 + x33 + x43 >= max { ( x1 + x2 + x3 + x4 ) , ( 1/x1 + 1/x2 + 1/x3 + 1/x4 ) }

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.