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The 14th Indian National Mathematical Olympiad
1999年第14届印度奥林匹克数学竞赛
  1. ABC is an acute-angled triangle. AD is an altitude, BE a median, and CF an angle bisector. CF meets AD at M, and DE at N. FM = 2, MN = 1, NC = 3. Find the perimeter of ABC.
  2. A rectangular field with integer sides and perimeter 3996 is divided into 1998 equal parts, each with integral area. Find the dimensions of the field.
  3. Show that x5 + 2x + 1 cannot be factorised into two polynomials with integer coefficients (and degree ≥ 1).
  4. X, X' are concentric circles. ABC, A'B'C' are equilateral triangles inscribed in X, X' respectively. P, P' are points on the perimeters of X, X' respectively. Show that P'A2 + P'B2 + P'C2 = A'P2 + B'P2 + C'P2.
  5. Given any four distinct reals, show that we can always choose three A, b, C, such that the equations ax2 + x + b = 0, bx2 + x + c = 0, cx2 + x + a = 0 either all have real roots, or all have non-real roots.
  6. For which n can {1, 2, 3, ... , 4n} be divided into n disjoint 4-element subsets such that for each subset one element is the arithmetic mean of the other three ?
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