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.