India National Olympiad 2010 Problems
Problem 1
Let $ ABC$ be a triangle with circum-circle $ \Gamma$. Let $ M$ be a point in the interior of triangle $ ABC$ which is also on the bisector of $ \angle A$. Let $ AM, BM, CM$ meet $ \Gamma$ in $ A_{1}, B_{1}, C_{1}$ respectively. Suppose $ P$ is the point of intersection of $ A_{1}C_{1}$ with $ AB$; and $ Q$ is the point of intersection of $ A_{1}B_{1}$ with $ AC$. Prove that $ PQ$ is parallel to $ BC$.
Problem 2
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n – 2)!$.
Problem 3
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 + xy + y^2)(y^2 + yz + z^2)(z^2 + zx + x^2) = xyz\]
\[ (x^4 + x^2y^2 + y^4)(y^4 + y^2z^2 + z^4)(z^4 + z^2x^2 + x^4) = x^3y^3z^3\]
Problem 4
How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions
\[ a_j^2 – a_ja_{j + 1} + a_{j + 1}^2\]
for $ j = 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?
Problem 5
Let $ ABC$ be an acute-angled triangle with altitude $ AK$. Let $ H$ be its ortho-centre and $ O$ be its circum-centre. Suppose $ KOH$ is an acute-angled triangle and $ P$ its circum-centre. Let $ Q$ be the reflection of $ P$ in the line $ HO$. Show that $ Q$ lies on the line joining the mid-points of $ AB$ and $ AC$.
Problem 6
Define a sequence $ < a_n > _{n\geq0}$ by $ a_0 = 0$, $ a_1 = 1$ and
\[ a_n = 2a_{n – 1} + a_{n – 2},\]
for $ n\geq2.$
$ (a)$ For every $ m > 0$ and $ 0\leq j\leq m,$ prove that $ 2a_m$ divides
$ a_{m + j} + ( – 1)^ja_{m – j}$.
$ (b)$ Suppose $ 2^k$ divides $ n$ for some natural numbers $ n$ and $ k$. Prove that $ 2^k$ divides $ a_n.$
