17th Iberoamerican 2002

17th Iberoamerican 2002 problems


A1. The numbers 1, 2, ... , 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ... , 3k+1th, ... numbers in the list are erased. Then the 1st, 4th, 7th, ... 3k+1th numbers in the remaining list are erased (leaving 3, 5, 8, 9, 12, ... ). This process is carried out repeatedly until there are no numbers left. What is the last number to be erased?
A2. Given a set of 9 points in the plane, no three collinear, show that for each point P in the set, the number of triangles containing P formed from the other 8 points in the set must be even.
A3. ABC is an equilateral triangle. P is a variable interior point such that angle APC = 120^o. The ray CP meets AB at M, and the ray AP meets BC at N. What is the locus of the circumcenter of the triangle MBN as P varies?
B1. ABC is a triangle. BD is the an angle bisector. E, F are the feet of the perpendiculars from A, C respectively to the line BD. M is the foot of the perpendicular from D to the line BC. Show that ∠DME = ∠DMF.
B2. The sequence a_n is defined as follows: a₁ = 56, a_n+1 = a_n - 1/a_n. Show that a_n < 0 for some n such that 0 < n < 2002.
B3. A game is played on a 2001 x 2001 board as follows. The first player's piece is the policeman, the second player's piece is the robber. Each piece can move one square south, one square east or one square northwest. In addition, the policeman (but not the robber) can move from the bottom right to the top left square in a single move. The policeman starts in the central square, and the robber starts one square diagonally northeast of the policeman. If the policeman moves onto the same square as the robber, then the robber is captured and the first player wins. However, the robber may move onto the same square as the policeman without being captured (and play continues). Show that the robber can avoid capture for at least 10000 moves, but that the policeman can ultimately capture the robber.

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