- Let ABC be a triangle and P be a interior point such that =, = . Let M,N be the mid-points of AC, BC respectively. Suppose BP=2PM. Prove A,P,N are collinear.
- Define a sequence as follows:
- if the number of positive divisors of n is odd.
- if the number of positive divisors of n is even. (The positive divisors of n include 1 as well as n.) Let be real number whose decimal expansion contains in the n-th place, . Determine with proof, whether is rational or irrational.

- Find all real number x such that . (Here [] denotes the largest integer not exceeding x.)
- All the points in the plane are coloured using three colours. Prove that there exists a triangle with vertices having the same colour such that either it is isosceles or its angles are in geometric progression.
- Let ABC be an acute-angled triangle and let H be its orthocentre. Let denotes the largest altitude of the triangle ABC. Prove that .
- Let a,b,c be positive real numbers such that. Prove that .