INMO 2006

In a non-equilateral triangle $A B C$, the sides $a, b, c$ form an arithmetic progression. Let $I$ and $O$ denote the incentre and circumcentre of the triangle respectively.

(i) Prove that $I O$ is perpendicular to $B I$.

(ii) Suppose $B I$ extended meets $A C$ in $K$, and $D, E$ are the midpoints of $B C, B A$ respectively. Prove that $I$ is the circumcentre of triangle $D K E$.

Prove that for every positive integer $n$ there exists a unique ordered pair $(a, b)$ of positive integers such that

$$n=\frac{1}{2}(a+b-1)(a+b-2)+a$$

Let $X$ denote the set of all triples $(a, b, c)$ of integers. Define a function $f: X \rightarrow X$ by

$$f(a, b, c)=(a+b+c, a b+b c+c a, a b c)$$

Find all triples $(a, b, c)$ in $X$ such that $f(f(a, b, c))=(a, b, c)$.

Some 46 squares are randomly chosen from a $9 \times 9$ chess board and are coloured red. Show that there exists a $2 \times 2$ block of 4 squares of which at least three are coloured red.

In a cyclic quadrilateral $A B C D, A B=a, B C=b, C D=c, \angle A B C=120^{\circ}$, and $\angle A B D=$ $30^{\circ}$. Prove that

(i) $c \geq a+b$;

(ii) $|\sqrt{c+a}-\sqrt{c+b}|=\sqrt{c-a-b}$.

(a) Prove that if $n$ is a positive integer such that $n \geq 4011^{2}$, then there exists an integer $l$ such that $n<l^{2}<\left(1+\frac{1}{2005}\right) n$.

(b) Find the smallest positive integer $M$ for which whenever an integer $n$ is such that $n \geq M$, there exists an integer $l$, such that $n<l^{2}<\left(1+\frac{1}{2005}\right) n$.