11th Brasil 1989


1. The triangle vertices (0,0), (0,1), (2,0) is repeatedly reflected in the three lines AB, BC, CA where A is (0,0), B is (3,0), C is (0,3). Show that one of the images has vertices (24,36), (24,37) and (26,36).
2. n is a positive integer such that n(n+1)/3 is a square. Show that n is a multiple of 3, and n+1 and n/3 are squares.
3. Let Z be the integers. f : Z → Z is defined by f(n) = n - 10 for n > 100 and f(n) = f(f(n+11)) for n ≤ 100. Find the set of possible values of f.
4. A and B play a game. Each has 10 tokens numbered from 1 to 10. The board is two rows of squares. The first row is numbered 1 to 1492 and the second row is numbered 1 to 1989. On the nth turn, A places his token number n on any empty square in either row and B places his token on any empty square in the other row. B wins if the order of the tokens is the same in the two rows, otherwise A wins. Which player has a winning strategy? Suppose each player has k tokens, numbered from 1 to k. Who has the winning strategy? Suppose that both rows are all the integers? Or both all the rationals?
5. The circumcenter of a tetrahedron lies inside the tetrahedron. Show that at least one of its edges is at least as long as the edge of a regular tetrahedron with the same circumsphere.

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