INMO 2009

Let $A B C$ be a triangle and let $P$ be an interior point such that $\angle B P C=90^{\circ}, \angle B A P=$ $\angle B C P$. Let $M, N$ be the mid-points of $A C, B C$ respectively. Suppose $B P=2 P M$. Prove that $A, P, N$ are collinear.

Define a sequence $\left\langle a_{n}\right\rangle_{n=1}^{\infty}$ as follows:

$$a_{n}= \begin{cases}0, & \text { if the number of positive divisors of } n \text { is odd } \\ 1, & \text { if the number of positive divisors of } n \text { is even }\end{cases}$$

(The positive divisors of $n$ include 1 as well as $n$.) Let $x=0 . a_{1} a_{2} a_{3} \ldots$ be the real number whose decimal expansion contains $a_{n}$ in the $n$-th place, $n \geq 1$. Determine, with proof, whether $x$ is rational or irrational.

Find all real numbers $x$ such that

$$\left[x^{2}+2 x\right]=[x]^{2}+2[x]$$

(Here $[x]$ 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 $A B C$ be an acute-angled triangle and let $H$ be its ortho-centre. Let $h_{\text {max }}$ denote the largest altitude of the triangle $A B C$. Prove that

$$A H+B H+C H \leq 2 h_{\max }$$

Let $a, b, c$ be positive real numbers such that $a^{3}+b^{3}=c^{3}$. Prove that

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