INMO 2007

In a triangle $A B C$ right-angled at $C$, the median through $B$ bisects the angle between $B A$ and the bisector of $\angle B$. Prove that

$$\frac{5}{2}<\frac{A B}{B C}<3$$

Let $n$ be a natural number such that $n=a^{2}+b^{2}+c^{2}$, for some natural numbers $a, b, c$. Prove that

$$9 n=\left(p_{1} a+q_{1} b+r_{1} c\right)^{2}+\left(p_{2} a+q_{2} b+r_{2} c\right)^{2}+\left(p_{3} a+q_{3} b+r_{3} c\right)^{2}$$

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 $9 n$ 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 .

Let $m$ and $n$ be positive integers such that the equation $x^{2}-m x+n=0$ 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 $x$.)

Let $\sigma=\left(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\right)$ be a permutation of $(1,2,3, \ldots, n)$. A pair $\left(a_{i}, a_{j}\right)$ is said to correspond to an inversion of $\sigma$, if $i<j$ but $a_{i}>a_{j}$. (Example: In the permutation $(2,4,5,3,1)$, there are 6 inversions corresponding to the pairs $(2,1)$, $(4,3),(4,1),(5,3),(5,1),(3,1)$.) How many permutations of $(1,2,3, \ldots n)$, $(n \geq 3)$, have exactly two inversions?

Let $A B C$ be a triangle in which $A B=A C$. Let $D$ be the mid-point of $B C$ and $P$ be a point on $A D$. Suppose $E$ is the foot of perpendicular from $P$ on $A C$. If $\frac{A P}{P D}=\frac{B P}{P E}=\lambda, \frac{B D}{A D}=m$ and $z=m^{2}(1+\lambda)$, prove that

$$z^{2}-\left(\lambda^{3}-\lambda^{2}-2\right) z+1=0$$

Hence show that $\lambda \geq 2$ and $\lambda=2$ if and only if $A B C$ is equilateral.

If $x, y, z$ are positive real numbers, prove that

$$(x+y+z)^{2}(y z+z x+x y)^{2} \leq 3\left(y^{2}+y z+z^{2}\right)\left(z^{2}+z x+x^{2}\right)\left(x^{2}+x y+y^{2}\right)$$