MathDB

1

Part of 2011 Serbia National Math Olympiad

Problems(2)

Easy sequence- Serbia Mathematical Olympiad 2011- Problem 1

Source:

4/8/2011
Let n≥2n \ge 2 be integer. Let a0a_0, a1a_1, ... ana_n be sequence of positive reals such that: (ak−1+ak)(ak+ak+1)=ak−1−ak+1(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}, for k=1,2,...,n−1k=1, 2, ..., n-1. Prove an<1n−1a_n< \frac{1}{n-1}.
inequalitiesinductionalgebra proposedalgebra
X, Y, O1, O2 are concyclic

Source: Serbia Math Olympiad 2011

4/8/2011
On sides AB,AC,BCAB, AC, BC are points M,X,YM, X, Y, respectively, such that AX=MXAX=MX; BY=MYBY=MY. KK, LL are midpoints of AYAY and BXBX. OO is circumcenter of ABCABC, O1O_1, O2O_2 are symmetric with OO with respect to KK and LL. Prove that X,Y,O1,O2X, Y, O_1, O_2 are concyclic.
geometrycircumcircleparallelogramgeometric transformationreflectiongeometry proposed