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The 19th Balkan Mathematical Olympiad
2002年第19届巴尔干地区数学奥林匹克
  1. Show that a finite graph in which every point has at least three edges contains an even cycle.
  2. The sequence an is defined by a1 = 20, a2 = 30, an+1 = 3an - an-1. Find all n for which 5an+1an + 1 is a square.
  3. Two unequal circles intersect at A and B. The two common tangents touch one circle at P, Q and the other at R, S. Show that the orthocenters of APQ, BPQ, ARS, BRS form a rectangle.
  4. N is the set of positive integers. Find all functions f: N → N
    such that f( f(n) ) + f(n) = 2n + 2001 or 2n + 2002.
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