India National Olympiad 2009 Problems
Problem 1
Let $ ABC$ be a tringle and let $ P$ be an interior point such that $ \angle BPC = 90 ,\angle BAP = \angle BCP$.Let $ M,N$ be the mid points of $ AC,BC$ respectively.Suppose $ BP = 2PM$.Prove that $ A,P,N$ are collinear.
Problem 2
Define a a sequence $ {<{a_n}>}^{\infty}_{n=1}$ as follows
$ a_n=0$, if number of positive divisors of $ n$ is odd
$ a_n=1$, if number of positive divisors of $ n$ is even
(The positive divisors of $ n$ include $ 1$ as well as $ n$.)Let $ x=0.a_1a_2a_3……..$ be the real number whose decimal expansion contains $ a_n$ in the $ n$-th place,$ n\geq1$.Determine,with proof,whether $ x$ is rational or irrational.
Problem 3
Find all real numbers $ x$ such that:
$ [x^2+2x]={[x]}^2+2[x]$
(Here $ [x]$ denotes the largest integer not exceeding $ x$.)
Problem 4
All the points in the plane are colored using three colors.Prove that there exists a triangle with vertices having the same color such that either it is isosceles or its angles are in geometric progression.
Problem 5
Let $ ABC$ be an acute angled triangle and let $ H$ be its ortho centre. Let $ h_{max}$ denote the largest altitude of the triangle $ ABC$. Prove that:
$AH + BH + CH\leq2h_{max}$
Problem 6
Let $ a,b,c$ be positive real numbers such that $ a^3 + b^3 = c^3$.Prove that:
$ a^2 + b^2 – c^2 > 6(c – a)(c – b)$.
