5th Iberoamerican 1990

5th Iberoamerican 1990 problems


A1. The function f is defined on the non-negative integers. f(2ⁿ - 1) = 0 for n = 0, 1, 2, ... . If m is not of the form 2ⁿ - 1, then f(m) = f(m+1) + 1. Show that f(n) + n = 2^k - 1 for some k, and find f(2¹⁹⁹⁰).
A2. I is the incenter of the triangle ABC and the incircle touches BC, CA, AB at D, E, F respectively. AD meets the incircle again at P. M is the midpoint of EF. Show that PMID is cyclic (or the points are collinear).
A3. f(x) = (x + b)² + c, where b and c are integers. If the prime p divides c, but p² does not divide c, show that f(n) is not divisible by p² for any integer n. If an odd prime q does not divide c, but divides f(n) for some n, show that for any r, we can find N such that q^r divides f(N).
B1. The circle C has diameter AB. The tangent at B is T. For each point M (not equal to A) on C there is a circle C' which touches T and touches C at M. Find the point at which C' touches T and find the locus of the center of C' as M varies. Show that there is a circle orthogonal to all the circles C'.
B2. A and B are opposite corners of an n x n board, divided into n² squares by lines parallel to the sides. In each square the diagonal parallel to AB is drawn, so that the board is divided into 2n² small triangles. The board has (n + 1)² nodes and large number of line segments, each of length 1 or √2. A piece moves from A to B along the line segments. It never moves along the same segment twice and its path includes exactly two sides of every small triangle on the board. For which n is this possible?
B3. f(x) is a polynomial of degree 3 with rational coefficients. If its graph touches the x-axis, show that it has three rational roots.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.