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.

Define a sequence \((a_n)_{n=1}^{\infty} \)as follows:

\(a_{n}=0 \)if the number of positive divisors of n is odd.

\(a_{n}=1 \) 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.

Find all real number x such that \([x^2+2x]=[x]^2+2[x] \). (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 \(h_{\max} \) denotes the largest altitude of the triangle ABC. Prove that \(AH+BH+CH \le 2h_{\max} \).

Let a,b,c be positive real numbers such that. Prove that \(a^2+b^2-c^2 > 6(c-a)(c-b) \).

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