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The 14th Balkan Mathematical Olympiad
1997年第14届巴尔干地区数学奥林匹克
  1. ABCD is a convex quadrilateral. X is a point inside it. XA2 + XB2 + XC2 + XD2 is twice the area of the quadrilateral. Show that it is a square and that X is its center.
  2. A collection of m subsets of X = {1, 2, ... , n} has the property that given any two elements of X we can find a subset in the collection which contains just one of the two. Prove that n ≤ 2m.
  3. Two circles C and C' lying outside each other touch at T. They lie inside a third circle and touch it at X and X' respectively. Their common tangent at T intersects the third circle at S. SX meets C again at P and XX' meets C again at Q. SX' meets C' again at U and XX' meets C' again at V. Prove that the lines ST, PQ and UV are concurrent.
  4. Find all real-valued functions on the reals which satisfy f( xf(x) + f(y) ) = f(x)2 + y for all x, y.
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