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I.S.I. and C.M.I. Entrance

INMO 2007

Try to solve these interesting INMO 2007 Questions. Solve them and write the answers in the comment to check your answers.

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 \( f(n)\neq 0\) .

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By Ashani Dasgupta

Founder Director at Cheenta
Pursuing Ph.D. in Mathematics from University of Wisconsin Milwaukee
Research Interest - Geometric Topology

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