Twenty-seventh International Olympiad, 1986

1986/1. Let d be any positive integer not equal to 2, 5 or 13. Show that one can find distinct a, b in the set {2, 5, 13, d} such that ab − 1 is not a perfect square.

1986/2. A triangle A1A2A3 and a point P0 are given in the plane. We define As= As−3 for all s ≥ 4. We construct a sequence of points P1, P2, P3, … such that Pk+1 is the image of Pk under rotation with center Ak+1 through angle 120° clockwise (for k = 0, 1, 2, …). Prove that if P1986 = P0, then the triangle A1A2A3 is equilateral.

1986/3. To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all the five numbers is positive. If three consecutive vertices are assigned the numbers x, y, z respectively and y < 0, then the following operation is allowed: the numbers x, y, z are replaced by x + y, −y, z + y respectively.

Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

1986/4. Let A, B be adjacent vertices of a regular n-gon (n ≥ 5) in the plane having center at O. A triangle XYZ, which is congruent to and initially coincides with OAB, moves in the plane in such a way that Y and Z each trace out the whole boundary of the polygon, X remaining inside the polygon. Find the locus of X.

1986/5. Find all functions f, defined on the nonnegative real numbers and taking nonnegative real values, such that:

(i) f(xf(y)) f(y) = f(x + y) for all x, y ≥ 0,

(ii) f(2) = 0,

(iii) f(x) ≠ 0 for 0 ≤ x < 2.

1986/6. One is given a finite set of points in the plane, each point having integer coordinates.