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The 12th Balkan Mathematical Olympiad
1995年第12届巴尔干地区数学奥林匹克
  1. Define an by a3 = (2 + 3)/(1 + 6), an = (an-1 + n)/(1 + n an-1). Find a1995.
  2. Two circles centers O and O' meet at A and B, so that OA is perpendicular to O'A. OO' meets the circles at C, E, D, F, so that the points C, O, E, D, O', F lie on the line in that order. BE meets the circle again at K and meets CA at M. BD meets the circle again at L and AF at N. Show that (KE/KM) (LN/LD) = (O'E/OD).
  3. m and n are positive integers with m > n and m + n even. Prove that the roots of x2 - (m2 - m + 1)(x - n2 - 1) - (n2 + 1)2 = 0 are positive integers, but not squares.
  4. Let S be an n x n array of lattice points. Let T be the set of all subsets of S of size 4 which form squares. Let A, B and C be the number of pairs {P, Q} of points of S which belong to respectively no, just two and just three elements of T. Show that A = B + 2C.
    [Note that there are plenty of squares tilted at an angle to the lattice and that the pair can be adjacent corners or opposite corners of the square.]
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