21st Brasil 1999


A1. ABCDE is a regular pentagon. The star ACEBD has area 1. AC and BE meet at P, BD and CE meet at Q. Find the area of APQD.
A2. Let d_n be the nth decimal digit of √2. Show that d_n cannot be zero for all of n = 1000001, 1000002, 1000003, ... , 3000000.
A3. How many pieces can be placed on a 10 x 10 board (each at the center of its square, at most one per square) so that no four pieces form a rectangle with sides parallel to the sides of the board?
B1. A spherical planet has finitely many towns. If there is a town at X, then there is also a town at X', the antipodal point. Some pairs of towns are connected by direct roads. No such roads cross (except at endpoints). If there is a direct road from A to B, then there is also a direct road from A' to B'. It is possible to get from any town to any other town by some sequence of roads. The populations of two towns linked by a direct road differ by at most 100. Show that there must be two antipodal towns whose populations differ by at most 100.
B2. n teams wish to play n(n-1)/2 games so that each team plays every other team just once. No team may play more than once per day. What is the minimum number of days required for the tournament?
B3. Given any triangle ABC, show how to construct A' on the side AB, B' on the side BC, C' on the side CA, so that ABC and A'B'C' are similar (with ∠A = ∠A', ∠B = ∠B' and ∠C = ∠C').

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