INMO 2009 Question Paper | Math Olympiad Problems

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:

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) $.

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