35th Spanish 1999


A1. A and B are points of the parabola y = x². The tangents at A and B meet at C. The median of the triangle ABC from C has length m. Find area ABC in terms of m.
A2. Show that there is an infinite sequence a₁, a₂, a₃, ... of positive integers such that a₁² + a₂² + ... + a_n² is a square for all n.
A3. A game is played on the board shown. A token is placed on each circle. Each token has a black side and a white side. Initially the topmost token has the black face showing, the others have the white face showing. A move is to remove token showing its black face and to turn over the tokens on the adjacent circles (joined by a line). Is it possible to remove all the tokens by a sequence of moves?
B1. A box contains 900 cards, labeled from 100 to 999. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that at least three of the digit sums of the cards removed are equal?
B2. G is the centroid of the triangle ABC. The distances of G from the three sides are g_a, g_b, g_c. Show that g_a ≥ 2r/3, and (g_a + g_b + g_c) ≥ 3r, where r is the inradius.
B3. Three families of parallel lines divide the plane into N regions. No three lines pass through the same point. What is the smallest number of lines needed to get N > 1999?

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