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INMO 2010

  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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