1. Let ABC be a triangle and P be a interior point such that \angle BPC =90^0 , \angle BAP = \angle BCP . Let M,N be the mid-points of AC, BC respectively. Suppose BP=2PM. Prove A,P,N are collinear.
  2. Define a sequence (a_n)_{n=1}^{\infty} as follows:
    1. a_{n}=0 if the number of positive divisors of n is odd.
    2. a_{n}=1 if the number of positive divisors of n is even. (The positive divisors of n include 1 as well as n.) Let x_n=0.a_1a_2a_3... be real number whose decimal expansion contains a_n in the n-th place, n ge 1 . Determine with proof, whether x is rational or irrational.
  3. Find all real number x such that [x^2+2x]=[x]^2+2[x] . (Here [] denotes the largest integer not exceeding x.)
  4. 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.
  5. Let ABC be an acute-angled triangle and let H be its orthocentre. Let h_{\max} denotes the largest altitude of the triangle ABC. Prove that AH+BH+CH \le 2h_{\max} .
  6. Let a,b,c be positive real numbers such that. Prove that a^2+b^2-c^2 > 6(c-a)(c-b) .