The 13th Indian National Mathematical Olympiad
1998年第13届印度奥林匹克数学竞赛 
 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 PA^{2} + PB^{2} = 4 PT^{2}.
 a, b are positive rationals such that a^{1/3} + b^{1/3} is also a rational. Show that a^{1/3} and b^{1/3} are rational.
 p, q, r, s are integers and s is not a multiple of 5. If there is an integer a such that pa^{3} + qa^{2} + ra + s is a multiple of 5, show that there is an integer b such that sb^{3} + rb^{2} + qb + p is a multiple of 5.
 ABCD is a cyclic quadrilateral inscribed in a circle radius 1. If AB·BC·CD·DA ≥ 4, show that ABCD is a square.
 The quadratic x^{2}  (a+b+c)x + (ab+bc+ca) = 0 has nonreal roots. Show that a, b, c, are all positive and that there is a triangle with sides √a, √b, √c.
 a_{1}, a_{2}, ... , a_{2n} is a sequence with two copies each of 0, 1, 2, ... , n1. 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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