10th Mexican 1996


A1. ABCD is a quadrilateral. P and Q are points on the diagonal BD such that the points are in the order B, P, Q, D and BP = PQ = QD. The line AP meets BC at E, and the line Q meets CD at F. Show that ABCD is a parallelogram iff E and F are the midpoints of their sides.
A2. 64 tokens are numbered 1, 2, ... , 64. The tokens are arranged in a circle around 1996 lamps which are all turned off. Each minute the tokens are all moved. Token number n is moved n places clockwise. More than one token is allowed to occupy the same place. After each move we count the number of tokens which occupy the same place as token 1 and turn on that number of lamps. Where is token 1 when the last lamp is turned on?
A3. Show that it is not possible to cover a 6 x 6 board with 1 x 2 dominos so that each of the 10 lines of length 6 that form the board (but do not lie along its border) bisects at least one domino. But show that we can cover a 5 x 6 board with 1 x 2 dominos so that each of the 9 lines of length 5 or 6 that form the board (but do not lie along its border) bisects at least one domino.
B1. For which n can we arrange the numbers 1, 2, 3, ... , 16 in a 4 x 4 array so that the eight row and column sums are all distinct and all multiples of n?
B2. Arrange the numbers 1, 2, 3, ... , n² in order in a n x n array (so that the first row is 1, 2, 3, ... , n, the second row is n+1, n+2, ... , 2n, and so on). For each path from 1 to n² which consists entirely of steps to the right and steps downwards, find the sum of the numbers in the path. Let M be the largest such sum and m the smallest. Show that M - m is a cube and that we cannot get the sum 1996 for a square of any size.
B3. ABC is an acute-angled triangle with AB < BC < AC. The points A', B', C' are such that AA' is perpendicular to BC and has the same length. Similarly, BB' is perpendicular to AC and has the same length, and CC' is perpendicular to AB and has the same length. The orthocenter H of ABC and A' are on the same side of A. Similarly, H and B' are on the same side of B, and H and C' are on the same side of C. Also ∠AC'B = 90^o. Show that A', B', C' are collinear.

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