Let be a non-equilateral triangle with integer sides. Let and be respectively the mid-points and ; let be the centroid of triangle . Suppose are concylic. Find the least possible perimeter of triangle .
For any natural number , consider a rectangular board made up of unit squares. This is covered by three types of tiles: tile, tile and domino. (For example, we can have types of tiling when : red-red,red-green, green-red, green-green and blue.) Let denote the number of ways of covering rectangular board by these three types of tiles. Prove that divides .
Let and be two circles with respective centres and intersecting in two distinct points and such that is an obtuse angle. Let the circumcircle of intersect and respectively in points and . Let the line intersect in ; let the line intersect in . Prove that, the points are concyclic.
Find all polynomials with real coefficients such that divides .
There are girls in a class sitting around a circular table, each having some apples with her. Every time the teacher notices a girl having more apples than both of her neighbours combined, the teacher takes away one apple from that girl and gives one apple each to her neighbours. Prove that, this process stops after a finite number of steps. (Assume that, the teacher has an abundant supply of apples.)
Let denote set of all natural numbers and let be a function such that
for all ;
divides for all .
Prove that, there exists an odd natural number such that for all in .