

A1. The function f(n) is defined on the positive integers and takes nonnegative integer values. f(2) = 0, f(3) > 0, f(9999) = 3333 and for all m, n:
f(m+n)  f(m)  f(n) = 0 or 1.
Determine f(1982).


A2. A nonisosceles triangle A_{1}A_{2}A_{3} has sides a_{1}, a_{2}, a_{3} with a_{i} opposite A_{i}. M_{i} is the midpoint of side a_{i} and T_{i} is the point where the incircle touches side a_{i}. Denote by S_{i} the reflection of T_{i} in the interior bisector of angle A_{i}. Prove that the lines M_{1}S_{1}, M_{2}S_{2} and M_{3}S_{3} are concurrent.


A3. Consider infinite sequences {x_{n}} of positive reals such that x_{0} = 1 and x_{0} ≥ x_{1} ≥ x_{2} ≥ ... .
(a) Prove that for every such sequence there is an n ≥ 1 such that:
x_{0}^{2}/x_{1} + x_{1}^{2}/x_{2} + ... + x_{n1}^{2}/x_{n} ≥ 3.999.
(b) Find such a sequence for which:
x_{0}^{2}/x_{1} + x_{1}^{2}/x_{2} + ... + x_{n1}^{2}/x_{n} < 4 for all n.


B1. Prove that if n is a positive integer such that the equation
x^{3}  3xy^{2} + y^{3} = n
has a solution in integers x, y, then it has at least three such solutions. Show that the equation has no solutions in integers for n = 2891.


B2. The diagonals AC and CE of the regular hexagon ABCDEF are divided by inner points M and N respectively, so that:
AM/AC = CN/CE = r.
Determine r if B, M and N are collinear.


B3. Let S be a square with sides length 100. Let L be a path within S which does not meet itself and which is composed of line segments A_{0}A_{1}, A_{1}A_{2}, A_{2}A_{3}, ... , A_{n1}A_{n} with A_{0} = A_{n}. Suppose that for every point P on the boundary of S there is a point of L at a distance from P no greater than 1/2. Prove that there are two points X and Y of L such that the distance between X and Y is not greater than 1 and the length of the part of L which lies between X and Y is not smaller than 198.

