12th BALKAN MATHEMATICAL OLYMPIAD
Plovdiv, Bulgaria, 1995
Find the value of the expression
where x * y = (x + y)/(1 + xy) for all positive x, y.
Consider two circles C1 and C2 with
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
C1 at K and intersects line AC at
M, and BD meets
C1 at K and intersects line AC at N. Show
r2 / r1 =
(KE / LM).(LN / ND)
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.
Let n be a positive integer and S the set of all points
(x, y) where x and y are positive integers
£ n, y £
n. Assume that T is the set of all squares whose vertices
belong to S.
Denote by ak
(k ³ 0) the number of pairs
of points in S which are the vertices of exactly k squares
from T. Prove that a0 = a2 +