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This post contains the problems from Indian National Mathematics Olympiad, INMO 2009 Question Paper. Do try to find their solutions.

**Indian National Mathematics Olympiad (INMO) 2009 Question Paper**:

- 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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