**1997
Mathematical Olympiad of Iran**
**First Day**

**April 24th,
1997**

**Time : 4 hours**

1. *x* and *y* are
two natural numbers such that *3x*^{2} + x =
4y^{2} +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. *x*_{1} ,* x*_{2}, *x*_{3}
and *x*_{4} are four positive real
numbers such that * x*_{1}x_{2}x_{3}x_{4} = 1 . Prove that :

*x*_{1}^{3} + x_{2}^{3} + x_{3}^{3} + x_{4}^{3} >= max { ( x_{1} + x_{2} + x_{3 }+ x_{4 }) , ( 1/x_{1} + 1/x_{2} + 1/x_{3 }+ 1/x_{4 })
}
5. In the triangle *ABC* ,
*B* and *C* are acute angles. The altitude of the
triangle drawn from *A*intersects *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.