INMO 2012

Let $A B C D$ be a quadrilateral inscribed in a circle. Suppose $A B=$ $\sqrt{2+\sqrt{2}}$ and $A B$ subtends $135^{\circ}$ at the centre of the circle. Find the maximum possible area of $A B C D$.

Let $p_{1}<p_{2}<p_{3}<p_{4}$ and $q_{1}<q_{2}<q_{3}<q_{4}$ be two sets of prime numbers such that $p_{4}-p_{1}=8$ and $q_{4}-q_{1}=8$. Suppose $p_{1}>5$ and $q_{1}>5$. Prove that 30 divides $p_{1}-q_{1}$.

Define a sequence $\left\langle f_{0}(x), f_{1}(x), f_{2}(x), \ldots\right\rangle$ of functions by

$$f_{0}(x)=1, f_{1}(x)=x, \left(f_{n}(x)\right)^{2}-1=f_{n+1}(x) f_{n-1}(x), \text { for } n \geq 1$$

Prove that each $f_{n}(x)$ is a polynomial with integer coefficients.

Let $A B C$ be a triangle. An interior point $P$ of $A B C$ is said to be good if we can find exactly 27 rays emanating from $P$ intersecting the sides of the triangle $A B C$ such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle $A B C$.

Let $A B C$ be an acute-angled triangle, and let $D, E, F$ be points on $B C$, $C A, A B$ respectively such that $A D$ is the median, $B E$ is the internal angle bisector and $C F$ is the altitude. Suppose $\angle F D E=\angle C, \angle D E F=$ $\angle A$ and $\angle E F D=\angle B$. Prove that $A B C$ is equilateral.

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function satisfying $f(0) \neq 0, f(1)=0$ and

(i) $f(x y)+f(x) f(y)=f(x)+f(y)$;

(ii) $(f(x-y)-f(0)) f(x) f(y)=0$,

for all $x, y \in \mathbb{Z}$, simultaneously.

(a) Find the set of all possible values of the function $f$.

(b) If $f(10) \neq 0$ and $f(2)=0$, find the set of all integers $n$ such that $f(n) \neq 0$.