MathDB

2

Part of 2011 Vietnam National Olympiad

Problems(2)

Show that the sequence has finite limits - [VMO 2011]

Source:

1/11/2011
Let ⟨xn⟩\langle x_n\rangle be a sequence of real numbers defined as x1=1;xn=2n(n−1)2∑i=1n−1xix_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i Show that the sequence yn=xn+1−xny_n=x_{n+1}-x_n has finite limits as n→∞.n\to \infty.
limitalgebra proposedalgebra
A, M, N, P are concyclic iff d passes through I- [VMO 2011]

Source:

1/12/2011
Let △ABC\triangle ABC be a triangle such that ∠C\angle C and ∠B\angle B are acute. Let DD be a variable point on BCBC such that D≠B,CD\neq B, C and ADAD is not perpendicular to BC.BC. Let dd be the line passing through DD and perpendicular to BC.BC. Assume d∩AB=E,d∩AC=F.d \cap AB= E, d \cap AC =F. If M,N,PM, N, P are the incentres of △AEF,△BDE,△CDF.\triangle AEF, \triangle BDE,\triangle CDF. Prove that A,M,N,PA, M, N, P are concyclic if and only if dd passes through the incentre of △ABC.\triangle ABC.
geometryincenterangle bisectorgeometry proposed