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INMO 2010 Questions - Indian National Mathematical Olympiad

This post contains Indian National Mathematical Olympiad, INMO 2010 questions. Try to solve these problems and share it in the comments.

  1. Let ABC be a triangle with circum-circle $ \Gamma $.Let M be a point in the interior of the triangle ABC which is also on the bisector of $ \angle A $. Let AM, BM, CM meets $ \Gamma $ in $ A_1,B_1,C_1 $ respectively. Suppose P is the point of intersection of $ A_1,B_1 $ with AC. Prove that PQ is parallel to BC.
  2. Find all natural numbers n>1 such that $ n^2 $ does not divide (n-2)!.
  3. Find all non-zero real numbers $ x,y,z $which satisfy the system of equations: $ (x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)=xyz $, $ x^4+x^2y^2+y^4)(y^4+y^2z^2+z^4)(z^4+z^2x^2+x^4)=x^3y^3z^3 $
  4. How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6) $are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6 $ is from the set {1,2,3,4} and the six expressions $ a_j^2-a_ja_{j+1}+a_{j+1}^2 $ for $ j=1,2,3,4,5,6 $ (where $ a_7 $ is to be taken as $ a_1 $ ) are all equal to one another?
  5. Let ABC be an acute-angled triangle with altitude AK. Let H be its orthocentre and O be its circumcentre. Suppose KOH is an acute-angled triangle and P its circumcircle. Let Q be the reflection of P in the line HO. Show that Q lies on the line joining the mid-points of AB and AC.
  6. Define a sequence $ (a_n)_{n \ge 0} $ by $ a_0=0 $ , $ a_1=1 $ and $ a_{n}=2a_{n-1}+a_{n-2} $, for $ n \ge 2 $. (a) For every $ m>0 $ and $ 0 \le j \le m $,prove that $ 2a_m $ divides $ a_{m+j}+(-1)^{j}a_{m-j} $.(b) Suppose $ 2^{k} $ divides n for some natural numbers n and k. Prove that $ 2^k $ divides a_n.

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