MathDB

2

Part of 2001 IMC

Problems(2)

easy group theory

Source: IMC 2001 day 1 problem 2

10/29/2005
Let r,s,tr,s,t positive integers which are relatively prime and a,b∈Ga,b \in G, GG a commutative multiplicative group with unit element ee, and ar=bs=(ab)t=ea^r=b^s=(ab)^t=e. (a) Prove that a=b=ea=b=e. (b) Does the same hold for a non-commutative group GG?
number theoryrelatively primesuperior algebrasuperior algebra unsolved
IMC 2001 Problem 8

Source: IMC 2001 Day 2 Problem 2

10/30/2020
Let a0=2,b0=2,an+1=2−4−an2,bn+1=2bn2+4+bn2a_{0}=\sqrt{2}, b_{0}=2,a_{n+1}=\sqrt{2-\sqrt{4-a_{n}^{2}}},b_{n+1}=\frac{2b_{n}}{2+\sqrt{4+b_{n}^{2}}}. a) Prove that the sequences (an)(a_{n}) and (bn)(b_{n}) are decreasing and converge to 00. b) Prove that the sequence (2nan)(2^{n}a_{n}) is increasing, the sequence (2nbn)(2^{n}b_{n}) is decreasing and both converge to the same limit. c) Prove that there exists a positive constant CC such that for all nn the following inequality holds: 0<bn−an<C8n0 <b_{n}-a_{n} <\frac{C}{8^{n}}.
inequalitiesConvergencereal analysis