Indian National Mathematics Olympiad 1995 Problems

Indian National Mathematics Olympiad 1995 Problems

1.  ABC is an acute-angled triangle with ∠A = 30^o. 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, ... , x_n are reals > 1 such that |x_i - x_i+1| < 1 for i < n. Show that x₁/x₂ + x₂/x₃ + ... + x_n-1/x_n + x_n/x₁ < 2n-1.

6.  Find all primes p for which (2^p-1 - 1)/p is a square.