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 $a^2+b^2-c^2 > 6(c-a)(c-b)$.