MathDB

2

Part of 2016 Vietnam National Olympiad

Problems(2)

two sequences

Source: Vietnam Mathematical Olympiad 2016, Day 1, Problem 2

1/6/2016
a) Let (an)(a_n) be the sequence defined by an=ln⁡(2n2+1)−ln⁡(n2+n+1)  ∀n≥1.a_n=\ln (2n^2+1)-\ln (n^2+n+1)\,\,\forall n\geq 1. Prove that the set {n∈N∣ {an}<12}\{n\in\mathbb{N}|\,\{a_n\}<\dfrac{1}{2}\} is a finite set; b) Let (bn)(b_n) be the sequence defined by an=ln⁡(2n2+1)+ln⁡(n2+n+1)  ∀n≥1a_n=\ln (2n^2+1)+\ln (n^2+n+1)\,\,\forall n\geq 1. Prove that the set {n∈N∣ {bn}<12016}\{n\in\mathbb{N}|\,\{b_n\}<\dfrac{1}{2016}\} is an infinite set.
algebralinear combination
Vietnam MO 2016 P6

Source: VMO 2016

7/29/2016
Given a triangle ABCABC inscribed by circumcircle (O)(O). The angles at B,CB,C are acute angle. Let MM on the arc BCBC that doesn't contain AA such that AMAM is not perpendicular to BCBC. AMAM meets the perpendicular bisector of BCBC at TT. The circumcircle (AOT)(AOT) meets (O)(O) at NN (N≠AN\ne A).
a) Prove that ∠BAM=∠CAN\angle{BAM}=\angle{CAN}.
b) Let II be the incenter and GG be the foor of the angle bisector of ∠BAC\angle{BAC}. AI,MI,NIAI,MI,NI intersect (O)(O) at D,E,FD,E,F respectively. Let P=DF∩AM,Q=DE∩AN{P}=DF\cap AM, {Q}=DE\cap AN. The circle passes through PP and touches ADAD at II meets DFDF at HH (H≠DH\ne D).The circle passes through QQ and touches ADAD at II meets DEDE at KK (K≠DK\ne D). Prove that the circumcircle (GHK)(GHK) touches BCBC.
geometryperpendicular bisectorcircumcircleincenterangle bisector