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The 3rd Balkan Mathematical Olympiad
1986年第3届巴尔干地区数学奥林匹克
  1. A line through the incenter of a triangle meets the circumcircle and incircle in the points A, B, C, D (in that order). Show that AB·CD ≥ BC2/4. When do you have equality?
  2. A point is chosen on each edge of a tetrahedron so that the product of the distances from the point to each end of the edge is the same for each of the 6 points. Show that the 6 points lie on a sphere.
  3. The integers r, s are non-zero and k is a positive real. The sequence an is defined by a1 = r, a2 = s, an+2 = (an+12 + k)/an. Show that all terms of the sequence are integers iff (r2 + s2 + k)/(rs) is an integer.
  4. A point P lies inside the triangle ABC and the triangles PAB, PBC, PCA all have the same area and the same perimeter. Show that the triangle is equilateral. If P lies outside the triangle, show that the triangle is right-angled.
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