MathDB

3

Part of 2001 Vietnam National Olympiad

Problems(2)

x_{n+1} = x_n + b \sin x_n

Source: 39-th Vietnamese Mathematical Olympiad 2001

3/19/2007
For real a,ba, b define the sequence x0,x1,x2,...x_{0}, x_{1}, x_{2}, ... by x0=a,xn+1=xn+bsin⁡xnx_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}. If b=1b = 1, show that the sequence converges to a finite limit for all aa. If b>2b > 2, show that the sequence diverges for some aa.
trigonometryintegrationcalculuscalculus computations
permutation of $\{1, 2, ... , 2n\}$

Source: 39-th Vietnamese Mathematical Olympiad 2001

3/19/2007
(a1,a2,...,a2n)(a_{1}, a_{2}, ... , a_{2n}) is a permutation of {1,2,...,2n}\{1, 2, ... , 2n\} such that ∣ai−ai+1∣≠∣aj−aj+1∣|a_{i}-a_{i+1}| \neq |a_{j}-a_{j+1}| for i≠ji \neq j. Show that a1=a2n+na_{1}= a_{2n}+n iff 1≤a2i≤n1 \leq a_{2i}\leq n for i=1,2,...n.i = 1, 2, ... n.
combinatorics unsolvedcombinatorics