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The 6th Balkan Mathematical Olympiad
1989年第6届巴尔干地区数学奥林匹克
  1. Find all integers which are the sum of the squares of their four smallest positive divisors.
  2. A prime p has decimal digits pnpn-1...p0 with pn > 1. Show that the polynomial pnxn + pn-1xn-1 + ... + p1x + p0 has no factors which are polynomials with integer coefficients and degree strictly between 0 and n.
  3. The triangle ABC has area 1. Take X on AB and Y on AC so that the centroid G is on the opposite of XY to B and C. Show that area BXGY + area CYGX ≥ 4/9. When do we have equality ?
  4. S is a collection of subsets of {1, 2, ... , n} of size 3. Any two distinct elements of S have at most one common element. Show that S cannot have more than n(n-1)/6 elements. Find a set S with n(n-4)/6 elements.
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