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INMO 2007

Try to solve these interesting INMO 2007 Questions.

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, of 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 the equation $x^2−mx+n = 0$ 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 $\sigma = (a_{1}, a_{2}, a_{3}, . . . , a_{n}$  be a permutation of  (1, 2, 3, . . . , n) . A pair     $a_{i}, a_{j}$ is said to correspond to an inversion of $\sigma$, if  < 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, . . . , n), (n>3) , have exactly two inversions.
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 the perpendicular from P on AC. If $\frac{AP}{PD}=\frac{BP}{PE}= \lambda$,$\frac{BD}{AD}= m$ and $z = m^2(1 + \lambda)$, prove that $z^2− (\sigma^3− \sigma^2− 2)z + 1 = 0$. Hence show that $\sigma ≥ 2$ and $\lambda = 2$ if and only if ABC is equilateral.
6. If x, y, z are positive real numbers, prove that $(x+y+z)^2(yz+zx+xy)^2≤ 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$.
7. Let  f :Z mapsto Z be a function satisfying $f(0) \neq 0$ , $f(1)=0$ and 1.f(xy) + f(x)f(y) = f(x) + f(y)\), 2. $(f(x-y) – f(0))f(x)f(y) = 0$ for all x , y in Z  simultaneously. 1. Find the set of all possible values of the function f. 2. If $f(10) \neq 0$ and  f(2) = 0 , find the set of all integers n such that $f(n)\neq 0$ .