MathDB

2

Part of 2013 Vietnam National Olympiad

Problems(2)

limit of squence

Source: Problem 2 VMO 2013 day 1

1/11/2013
Define a sequence {an}\{a_n\} as: {a1=1an+1=3−an+22an  for n≥1.\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.
Prove that this sequence has a finite limit as n→+∞n\to+\infty . Also determine the limit.
limitfunctionalgebra proposedalgebra
fixed circle

Source: Problem 6 VMO 2013 day 2

1/12/2013
Let ABCABC be a cute triangle.(O)(O) is circumcircle of △ABC\triangle ABC.DD is on arc BCBC not containing AA.Line △\triangle moved through HH(HH is orthocenter of △ABC\triangle ABC cuts circumcircle of △ABH\triangle ABH,circumcircle △ACH\triangle ACH again at M,NM,N respectively. a.Find △\triangle satisfy SAMNS_{AMN} max b.d1,d2d_{1},d_{2} are the line through MM perpendicular to DBDB,the line through NN perpendicular to DCDC respectively. d1d_{1} cuts d2d_{2} at PP.Prove that PP move on a fixed circle.
geometrycircumcirclegeometry proposed