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The 1st Balkan Mathematical Olympiad
1984年第1届巴尔干地区数学奥林匹克
  1. Let x1, x2, ... , xn be positive reals with sum 1. Prove that x1/(2 - x1) + x2/(2 - x2) + ... + xn/(2 - xn) ≥ n/(2n - 1).
  2. ABCD is a cyclic quadrilateral. A' is the orthocenter (point where the altitudes meet) of BCD, B' is the orthocenter of ACD, C' is the orthocenter of ABD, and D' is the orthocenter of ABC. Prove that ABCD and A'B'C'D' are congruent.
  3. Prove that given any positive integer n we can find a larger integer m such that the decimal expansion of 5m can be obtained from that for 5n by adding additional digits on the left.
  4. Given positive reals a, b, c find all real solutions (x, y, z) to the equations
    ax + by = (x - y)2, by + cz = (y - z)2, cz + ax = (z - x)2.
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