# [Solutions] India National Mathematical Olympiad 2007

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 $\frac{5}{2} < \frac{AB}{BC} < 3$
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 $3$.
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 $x$)
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 $(2,1),(4,3),(4,1),(5,3),(5,1),(3,1)$.
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.
6. If $x$, $y$, $z$ are positive real numbers. Prove that $\left(x + y + z\right)^2 \left(yz + zx + xy\right)^2 \leq 3\left(y^2 + yz + z^2\right)\left(z^2 + zx + x^2\right)\left(x^2 + xy + y^2\right) .$
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