India National Olympiad 2012 Problems
Problem 1
Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.
Problem 2
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 $p_1 – q_1$.
Problem 3
Define a sequence $<f_0 (x), f_1 (x), f_2 (x), \dots>$ of functions by$$f_0 (x) = 1$$$$f_1(x)=x$$$$(f_n(x))^2 – 1 = f_{n+1}(x) f_{n-1}(x)$$for $n \ge 1$. Prove that each $f_n (x)$ is a polynomial with integer coefficients.
Problem 4
Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be good if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ 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 $ABC$.
Problem 5
Let $ABC$ be an acute angled triangle. Let $D,E,F$ be points on $BC, CA, AB$ such that $AD$ is the median, $BE$ is the internal bisector and $CF$ is the altitude. Suppose that $\angle FDE=\angle C, \angle DEF=\angle A$ and $\angle EFD=\angle B.$ Show that $ABC$ is equilateral.
Problem 6
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and
$(i) f(xy) + f(x)f(y) = f(x) + f(y)$
$(ii)\left(f(x-y) – f(0)\right ) 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 $f(10) \ne 0$ and $f(2) = 0$, find the set of all integers $n$ such that $f(n) \ne 0$.
