Indian National Mathematics Olympiad 1998 Problems

Indian National Mathematics Olympiad 1998 Problems

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 a^1/3 + b^1/3 is also a rational. Show that a^1/3 and b^1/3 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, ... , a_2n 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.