Let ABC be a non-equilateral triangle with integer sides. Let D and E be respectively the mid-points BC and C A, let G be the centroid of triangle ABC. Suppose D, C, E, G are concyclic. Find the least possible perimeter of triangle ABC.

For any natural number n, consider a \(1 \times n\) rectangular board made up of n unit squares. This is covered by three types of tiles; \(1\times 1\) red tile, \(1\times 1\) green tile and \(1\times 2\) blue domino. (For example, we can have 5 types of tiling when n = 2; red-red; red-green; green-red; green-green; and blue.) Let \(t_n\) denote the number of ways of covering \(1\times n\) rectangular board by these three types of tiles. Prove that \(t_n\) divides \(t_{2n+1}\).

Let \(\Gamma_1\) and \(\Gamma_2\) be two circles with respective centres \(O_1\) and \(O_2\) intersecting in two distinct points A and B such that \({\angle O_1}A{O_2}\) is an obtuse angle. Let the circumcircle of triangle \({O_1}A{O_2}\) intersect \(\Gamma_1\) \(T_2\) respectively in points \(C{(\not= A)}\) and \(D{(\not= A)}\). Let the line C B intersect \(\Gamma_2\) in E; let the line D B intersect \(\Gamma_1\) in F . Prove that the points C, D, E, F are concyclic.

Find all polynomials with real coefficients P(x) such that \(P{(x^2+x+1)}\) divides \(P{(x^3-1)}\).

There are \(n\ge 3\) 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 neighnors combined, the teacher takes away one apple from that girl and gives one apple each to her neighbors. Prove that this process stops after a finite numberof steps. (Assume that the teacher has an abundant supply of apples.)

Let N denote the set of all natural numbers and let \(f : N\rightarrow N\) be a function such that
(a) \(f{(mn)} = f {(m)} f{(n)}\) for all m,n in N ;
(b) m+n divides \(f {(m)} + f {(n)} \) for all m, n in N
Prove that there exists an odd natural number \(k\) such that \(f {(n)} = n^k\) for all n in N.