MathDB

1

Part of 1998 Vietnam National Olympiad

Problems(2)

ln and sequence

Source: 36-th Vietnamese Mathematical Olympiad 1998

2/16/2007
Let a≥1a\geq 1 be a real number. Put x1=a,xn+1=1+ln⁡(xn21+ln⁡xn)(n=1,2,...)x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...). Prove that the sequence {xn}\{x_{n}\} converges and find its limit.
logarithmsintegrationlimitreal analysisreal analysis unsolved
Does there exist an infinite sequence ...?

Source: 36-th Vietnamese Mathematical Olympiad 1998

2/16/2007
Does there exist an infinite sequence {xn}\{x_{n}\} of reals satisfying the following conditions i)∣xn∣≤0,666|x_{n}|\leq 0,666 for all n=1,2,...n=1,2,... ii)∣xm−xn∣≥1n(n+1)+1m(m+1)|x_{m}-x_{n}|\geq \frac{1}{n(n+1)}+\frac{1}{m(m+1)} for all m≠nm\not = n?
algebra proposedalgebra