| A1. Chords AB and CD of a circle intersect at a point E inside the circle. Let M be an interior point of the segment EB. The tangent at E to the circle through D, E and M intersects the lines BC and AC at F and G respectively. Find EF/EG in terms of t = AM/AB. |
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| A2. Take n ≥ 3 and consider a set E of 2n-1 distinct points on a circle. Suppose that exactly k of these points are to be colored black. Such a coloring is "good" if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly n points from E. Find the smallest value of k so that every such coloring of k points of E is good. |
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| A3. Determine all integers greater than 1 such that (2n + 1)/n2 is an integer. |
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| B1. Construct a function from the set of positive rational numbers into itself such that f(x f(y)) = f(x)/y for all x, y. |
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B2. Given an initial integer n0 > 1, two players A and B choose integers n1, n2, n3, ... alternately according to the following rules:
Knowing n2k, A chooses any integer n2k+1 such that n2k ≤ n2k+1 ≤ n2k2. Knowing n2k+1, B chooses any integer n2k+2 such that n2k+1/n2k+2 = pr for some prime p and integer r ≥ 1. Player A wins the game by choosing the number 1990; player B wins by choosing the number 1. For which n0 does
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| B3. Prove that there exists a convex 1990-gon such that all its angles are equal and the lengths of the sides are the numbers 12, 22, ... , 19902 in some order. |
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