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The 10th Indian National Mathematical Olympiad
1995年第10届印度奥林匹克数学竞赛
  1. ABC is an acute-angled triangle with ∠A = 30o. H is the orthocenter and M is the midpoint of BC. T is a point on HM such that HM = MT. Show that AT = 2 BC.
  2. Show that there are infinitely many pairs (a,b) of coprime integers (which may be negative, but not zero) such that x2 + ax + b = 0 and x2 + 2ax + b have integral roots.
  3. Show that more 3 element subsets of {1, 2, 3, ... , 63} have sum greater than 95 than have sum less than 95.
  4. ABC is a triangle with incircle K, radius r. A circle K', radius r', lies inside ABC and touches AB and AC and touches K externally. Show that r'/r = tan2((π-A)/4).
  5. x1, x2, ... , xn are reals > 1 such that |xi - xi+1| < 1 for i < n. Show that x1/x2 + x2/x3 + ... + xn-1/xn + xn/x1 < 2n-1.
  6. Find all primes p for which (2p-1 - 1)/p is a square.
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