India National Olympiad 2011 Problems
Problem 1
Let $D,E,F$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE.$ Show that $\triangle ABC$ is equilateral.
Problem 2
Call a natural number $n$ faithful if there exist natural numbers $a<b<c$ such that $a|b,$ and $b|c$ 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.
Problem 3
Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$
Problem 4
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.
Problem 5
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Gamma.$ Let $E,F,G,H$ be the midpoints of arcs $AB,BC,CD,AD$ of $\Gamma,$ respectively. Suppose that $AC\cdot BD=EG\cdot FH.$ Show that $AC,BD,EG,FH$ are all concurrent.
Problem 6
Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying
\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]For all $x,y\in\mathbb R$.
