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 nnumber.
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
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.
meoTxe amocana ar daido ise, rogorc saWiroa, bevri Secdoma iyo, amitom saerTod a idos is jobia. tu vtqvaT mainc Zalian gainteresebT, SegiZliaT aq naxoT: http://www.geocities.com/CollegePark/Lounge/5284/math/
