14th Iberoamerican Mathematical Olympiad Problems 1999

A1.  Find all positive integers n < 1000 such that the cube of the sum of the digits of n equals n².

A2.  Given two circles C and C' we say that C bisects C' if their common chord is a diameter of C'. Show that for any two circles which are not concentric, there are infinitely many circles which bisect them both. Find the locus of the centers of the bisecting circles.

A3.  Given points P₁, P₂, ... , P_n on a line we construct a circle on diameter P_iP_j for each pair i, j and we color the circle with one of k colors. For each k, find all n for which we can always find two circles of the same color with a common external tangent.

B1.  Show that any integer greater than 10 whose digits are all members of {1, 3, 7, 9} has a prime factor ≥ 11.

B2.  O is the circumcenter of the acute-angled triangle ABC. The altitudes are AD, BE and CF. The line EF cuts the circumcircle at P and Q. Show that OA is perpendicular to PQ. If M is the midpoint of BC, show that AP² = 2 AD·OM.

B3.  Given two points A and B, take C on the perpendicular bisector of AB. Define the sequence C₁, C₂, C₃, ... as follows. C₁ = C. If C_n is not on AB, then C_n+1 is the circumcenter of the triangle ABC_n. If C_n lies on AB, then C_n+1 is not defined and the sequence terminates. Find all points C such that the sequence is periodic from some point on.