The 8th Balkan Mathematical Olympiad
1991年第8届巴尔干地区数学奥林匹克 |
- The circumcircle of the acute-angled triangle ABC has center O. M lies on the minor arc AB. The line through M perpendicular to OA cuts AB at K and AC at L. The line through M perpendicular to OB cuts AB at N and BC at P. MN = KL. Find angle MLP in terms of angles A, B and C.
- Find an infinite set of incongruent triangles each of which has integral area and sides which are relatively prime integers, but none of whose altitudes are integral.
- A regular hexagon area H has its vertices on the perimeter of a convex polygon of area A. Prove that 2A ≤ 3H. When do we have equality ?
- A is the set of positive integers and B is A ∪ {0}. Prove that no bijection f: A → B can satisfy f(mn) = f(m) + f(n) + 3 f(m) f(n) for all m, n.
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