31st International Mathematical Olympiad 1990 Problems & Solutions

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. A2.  Take n ≥ 3 and consider a set E of 2n-1 distinct...

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30th International Mathematical Olympiad 1989 Problems & Solutions

A1.  Prove that the set {1, 2, ... , 1989} can be expressed as the disjoint union of subsets A1, A2, ... , A117 in such a way that each Ai contains 17 elements and the sum of the elements in each Ai is the same. A2.  In an acute-angled triangle ABC, the internal bisector of angle A meets the circumcircle again at...

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29th International Mathematical Olympiad 1988 Problems & Solutions

A1.  Consider two coplanar circles of radii R > r with the same center. Let P be a fixed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C. The perpendicular to BP at P meets the smaller circle again at A (if it is tangent to the circle at P, then A = P). ...

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28th International Mathematical Olympiad 1987 Problems & Solutions

A1.  Let pn(k) be the number of permutations of the set {1, 2, 3, ... , n} which have exactly k fixed points. Prove that the sum from k = 0 to n of (k pn(k) ) is n!. [A permutation f of a set S is a one-to-one mapping of S onto itself. An element i of S is called a fixed point if f(i) = i.] A2.  In an acute-angled...

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27th International Mathematical Olympiad 1986 Problems & Solutions

A1.  Let d be any positive integer not equal to 2, 5 or 13. Show that one can find distinct a, b in the set {2, 5, 13, d} such that ab - 1 is not a perfect square. A2.  Given a point P0 in the plane of the triangle A1A2A3. Define As = As-3 for all s >= 4. Construct a set of points P1, P2, P3, ... such that Pk+1...

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