23rd Brasil 2001


A1. Prove that (a + b)(a + c) ≥ 2( abc(a + b + c) )^1/2 for all positive reals.
A2. Given a₀ > 1, the sequence a₀, a₁, a₂, ... is such that for all k > 0, a_k is the smallest integer greater than a_k-1 which is relatively prime to all the earlier terms in the sequence. Find all a₀ for which all terms of the sequence are primes or prime powers.
A3. ABC is a triangle. The points E and F divide AB into thirds, so that AE = EF = FB. D is the foot of the perpendicular from E to the line BC, and the lines AD and CF are perpendicular. ∠ACF = 3 ∠BDF. Find DB/DC.
B1. A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)
B2. An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.
B3. A one-player game is played as follows. There is bowl at each integer on the x-axis. All the bowls are initially empty, except for that at the origin, which contains n stones. A move is either (A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or (B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left. Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for given n).

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