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The 20th Balkan Mathematical Olympiad
2003年第20届巴尔干地区数学奥林匹克
  1. Is there a set of 4004 positive integers such that the sum of each subset of 2003 elements is not divisible by 2003 ?
  2. ABC is a triangle. The tangent to the circumcircle at A meets the line BC at D. The perpendicular to BC at B meets the perpendicular bisector of AB at E, and the perpendicular to BC at C meets the perpendicular bisector of AC at F. Show that D, E, F are collinear.
  3. Find all real-valued functions f(x) on the rationals such that:
    (1) f(x + y) - y f(x) - x f(y) = f(x) f(y) - x - y + xy, for all x, y
    (2) f(x) = 2 f(x+1) + 2 + x, for all x and
    (3) f(1) + 1 > 0.
  4. A rectangle ABCD has side lengths AB = m, AD = n, with m and n relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points A1 = A, A2, A3, ... , AN = C. Show that A1A2 - A2A3 + A3A4 - ... ± AN-1AN = AC/(mn).
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