12th Iberoamerican Mathematical Olympiad Problems 1997

A1.  k ≥ 1 is a real number such that if m is a multiple of n, then [mk] is a multiple of [nk]. Show that k is an integer.

A2.  I is the incenter of the triangle ABC. A circle with center I meets the side BC at D and P, with D nearer to B. Similarly, it meets the side CA at E and Q, with E nearer to C, and it meets AB at F and R, with F nearer to A. The lines EF and QR meet at S, the lines FD and RP meet at T, and the lines DE and PQ meet at U. Show that the circumcircles of DUP, ESQ and FTR have a single point in common.

A3.  n > 1 is an integer. D_n is the set of lattice points (x, y) with |x|, |y| ≤ n. If the points of D_n are colored with three colors (one for each point), show that there are always two points with the same color such that the line containing them does not contain any other points of D_n. Show that it is possible to color the points of D_n with four colors (one for each point) so that if any line contains just two points of D_n then those two points have different colors.

B1.  Let o(n) be the number of 2n-tuples (a₁, a₂, ... , a_n, b₁, b₂, ... , b_n) such that each a_i, b_j = 0 or 1 and a₁b₁ + a₂b₂ + ... + a_nb_n is odd. Similarly, let e(n) be the number for which the sum is even. Show that o(n)/e(n) = (2ⁿ - 1)/(2ⁿ + 1).

B2.  ABC is an acute-angled triangle with orthocenter H. AE and BF are altitudes. AE is reflected in the angle bisector of angle A and BF is reflected in the angle bisector of angle B. The two reflections intersect at O. The rays AE and AO meet the circumcircle of ABC at M and N respectively. P is the intersection of BC and HN, R is the intersection of BC and OM, and S is the intersection of HR and OP. Show that AHSO is a parallelogram.

B3.  Given 1997 points inside a circle of radius 1, one of them the center of the circle. For each point take the distance to the closest (distinct) point. Show that the sum of the squares of the resulting distances is at most 9.