### 12^{th} BALKAN MATHEMATICAL OLYMPIAD
Plovdiv, Bulgaria, 1995

### Problem 1

Find the value of the expression

(...(((2*3)*4*5)*...)*1995,

where x * y = (x + y)/(1 + xy) for all positive x, y.
### Problem 2

Consider two circles *C*_{1} and *C*_{2} with
centers
*O*_{1}, *O*_{2} and radii
*r*_{1}, *r*_{2}, respectively
(*r*_{2} > *r*_{1}) which intersect
at *A* and *B* such that
*O*_{1}*AO*_{2} = 90°.
The line *O*_{1}*O*_{2} intersects
*C*_{1} at *C*, *D* and *C*_{2} at
*E*, *F* where *E* lies between *C* and *D* and
*D* lies between *E* and *F*. Line *BE*
meets
*C*_{1} at *K* and intersects line *AC* at
*M*, and *BD* meets
*C*_{1} at K and intersects line *AC* at *N*. Show
that

*r*_{2} / *r*_{1} =
(*KE* / *LM*).(*LN* / *ND*)
### Problem 3

Let *a*, *b* be positive integers such that *a* > *b*
and *a* + *b* is even.
Prove that the roots of the equation
*x*^{2} - (*a*^{2} - *a* + 1)(*x* -
*b*^{2} - 1)
- (*b*^{2} + 1)^{2} = 0
are positive integers, none of which is a perfect square.
### Problem 4

Let *n* be a positive integer and *S* the set of all points
(*x*, *y*) where *x* and *y* are positive integers
with *x*
£ *n*, *y* £
*n*. Assume that *T* is the set of all squares whose vertices
belong to *S*.
Denote by *a*_{k}
(*k* ³ 0) the number of pairs
of points in *S* which are the vertices of exactly *k* squares
from *T*. Prove that *a*_{0} = *a*_{2} +
2 *a*_{3}.