India National Olympiad 2007 Problems
Problem 1
In a triangle $ ABC$ right-angled at $ C$ , the median through $ B$ bisects the angle between $ BA$ and the bisector of $ \angle B$. Prove that
\[ \frac{5}{2} < \frac{AB}{BC} < 3\]
Problem 2
Let $ n$ be a natural number such that $ n = a^2 + b^2 +c^2$ for some natural numbers $ a,b,c$. Prove that
\[ 9n = (p_1a+q_1b+r_1c)^2 + (p_2a+q_2b+r_2c)^2 + (p_3a+q_3b+r_3c)^2\]
where $ p_j$’s , $ q_j$’s , $ r_j$’s are all nonzero integers. Further, if $ 3$ does not divide at least one of $ a,b,c,$ prove that $ 9n$ can be expressed in the form $ x^2+y^2+z^2$, where $ x,y,z$ are natural numbers none of which is divisible by $ 3$.
Problem 3
Let $ m$ and $ n$ be positive integers such that $ x^2 – mx +n = 0$ has real roots $ \alpha$ and $ \beta$.
Prove that $ \alpha$ and $ \beta$ are integers if and only if $ [m\alpha] + [m\beta]$ is the square of an integer.
(Here $ [x]$ denotes the largest integer not exceeding $ x$)
Problem 4
Let $ \sigma = (a_1, a_2, \cdots , a_n)$ be permutation of $ (1, 2 ,\cdots, n)$. A pair $ (a_i, a_j)$ is said to correspond to an inversion of $\sigma$ if $ i<j$ but $ a_i>a_j$. How many permutations of $ (1,2,\cdots,n)$, $ n \ge 3$, have exactly two inversions?
For example, In the permutation $(2,4,5,3,1)$, there are 6 inversions corresponding to the pairs $ (2,1),(4,3),(4,1),(5,3),(5,1),(3,1)$.
Problem 5
Let $ ABC$ be a triangle in which $ AB=AC$. Let $ D$ be the midpoint of $ BC$ and $ P$ be a point on $ AD$. Suppose $ E$ is the foot of perpendicular from $ P$ on $ AC$. Define
\[ \frac{AP}{PD}=\frac{BP}{PE}=\lambda , \ \ \ \frac{BD}{AD}=m , \ \ \ z=m^2(1+\lambda)\]
Prove that
\[ z^2 – (\lambda^3 – \lambda^2 – 2)z + 1 = 0\]
Hence show that $ \lambda \ge 2$ and $ \lambda = 2$ if and only if $ ABC$ is equilateral.
Problem 6
If $ x$, $ y$, $ z$ are positive real numbers, prove that
\[ \left(x + y + z\right)^2 \left(yz + zx + xy\right)^2 \leq 3\left(y^2 + yz + z^2\right)\left(z^2 + zx + x^2\right)\left(x^2 + xy + y^2\right) .\]
