# INMO 2009

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