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The 13th Balkan Mathematical Olympiad
1996年第13届巴尔干地区数学奥林匹克
  1. Let d be the distance between the circumcenter and the centroid of a triangle. Let R be its circumradius and r the radius of its inscribed circle. Show that d2 ≤ R(R - 2r).
  2. p > 5 is prime. A = {p - n2 where n2 < p}. Show that we can find integers a and b in A such that a > 1 and a divides b.
  3. In a convex pentagon consider the five lines joining a vertex to the midpoint of the opposite side. Show that if four of these lines pass through a point, then so does the fifth.
  4. Can we find a subset X of {1, 2, 3, ... , 21996-1} with at most 2012 elements such that 1 and 21996-1 belong to X and every element of X except 1 is the sum of two distinct elements of X or twice an element of X ?
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