INMO 2011

Let $D, E, F$ be points on the sides $B C, C A, A B$ respectively of a triangle $A B C$ such that $B D=C E=A F$ and $\angle B D F=\angle C E D=\angle A F E$. Prove that $A B C$ is equilateral.

Call a natural number $n$ faithful, if there exist natural numbers $a<b<c$ such that $a$ divides $b, b$ divides $c$ and $n=a+b+c$.

(i) Show that all but a finite number of natural numbers are faithful.

(ii) Find the sum of all natural numbers which are not faithful.

Consider two polynomials $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ and $Q(x)=b_{n} x^{n}+$ $b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0}$ with integer coefficients such that $a_{n}-b_{n}$ is a prime, $a_{n-1}=b_{n-1}$ and $a_{n} b_{0}-a_{0} b_{n} \neq 0$. Suppose there exists a rational number $r$ such that $P(r)=Q(r)=0$. Prove that $r$ is an integer.

Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.

Let $A B C D$ be a quadrilateral inscribed in a circle $\Gamma$. Let $E, F, G, H$ be the midpoints of the arcs $A B, B C, C D, D A$ of the circle $\Gamma$. Suppose $A C \cdot B D=E G \cdot F H$. Prove that $A C, B D$, $E G, F H$ are concurrent.

Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that

$$f(x+y) f(x-y)=(f(x)+f(y))^{2}-4 x^{2} f(y)$$

for all $x, y \in \mathbf{R}$, where $\mathbf{R}$ denotes the set of all real numbers.