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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.