23rd IMO 1982

A1.  The function f(n) is defined on the positive integers and takes non-negative 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 non-isosceles triangle A1A2A3 has sides a1, a2, a3 with ai opposite Ai. Mi is the midpoint of side ai and Ti is the point where the incircle touches side ai. Denote by Si the reflection of Ti in the interior bisector of angle Ai. Prove that the lines M1S1, M2S2 and M3S3 are concurrent.
A3.  Consider infinite sequences {xn} of positive reals such that x0 = 1 and x0 ≥ x1 ≥ x2 ≥ ... .

(a)  Prove that for every such sequence there is an n ≥ 1 such that:

      x02/x1 + x12/x2 + ... + xn-12/xn ≥ 3.999.

(b)  Find such a sequence for which:

      x02/x1 + x12/x2 + ... + xn-12/xn < 4   for all n.

B1.  Prove that if n is a positive integer such that the equation

      x3 - 3xy2 + y3 = 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 A0A1, A1A2, A2A3, ... , An-1An with A0 = An. 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.
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John Scholes
14 Oct 1998