12th 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 C1 and C2 with centers O1, O2 and radii r1, r2, respectively (r2 > r1) which intersect at A and B such that O1AO2 = 90. The line O1O2 intersects C1 at C, D and C2 at E, F where E lies between C and D and D lies between E and F. Line BE meets C1 at K and intersects line AC at M, and BD meets C1 at K and intersects line AC at N. Show that

r2 / r1 = (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
x2 - (a2 - a + 1)(x - b2 - 1) - (b2 + 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 ak (k 0) the number of pairs of points in S which are the vertices of exactly k squares from T. Prove that a0 = a2 + 2 a3.