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The 13th Indian National Mathematical Olympiad
1998年第13届印度奥林匹克数学竞赛
  1. C is a circle with center O. AB is a chord not passing through O. M is the midpoint of AB. C' is the circle diameter OM. T is a point on C'. The tangent to C' at T meets C at P. Show that PA2 + PB2 = 4 PT2.
  2. a, b are positive rationals such that a1/3 + b1/3 is also a rational. Show that a1/3 and b1/3 are rational.
  3. p, q, r, s are integers and s is not a multiple of 5. If there is an integer a such that pa3 + qa2 + ra + s is a multiple of 5, show that there is an integer b such that sb3 + rb2 + qb + p is a multiple of 5.
  4. ABCD is a cyclic quadrilateral inscribed in a circle radius 1. If AB·BC·CD·DA ≥ 4, show that ABCD is a square.
  5. The quadratic x2 - (a+b+c)x + (ab+bc+ca) = 0 has non-real roots. Show that a, b, c, are all positive and that there is a triangle with sides √a, √b, √c.
  6. a1, a2, ... , a2n is a sequence with two copies each of 0, 1, 2, ... , n-1. A subsequence of n elements is chosen so that its arithmetic mean is integral and as small as possible. Find this minimum value.
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