- Let ABC be a triangle with circum-circle .Let M be a point in the interior of the triangle ABC which is also on the bisector of . Let AM, BM, CM meets in respectively. Suppose P is the point of intersection of with AC. Prove that PQ is parallel to BC.
- Find all natural numbers n>1 such that does not divide (n-2)!.
- Find all non-zero real numbers which satisfy the system of equations: ,
- How many 6-tuples are there such that each of is from the set {1,2,3,4} and the six expressions for (where is to be taken as ) are all equal to one another?
- 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.
- Define a sequence by , and , for . (a) For every and ,prove that divides .(b) Suppose divides n for some natural numbers n and k. Prove that divides a_n.