MathDB

3

Part of 1994 Vietnam National Olympiad

Problems(2)

x_{n+1}=f(x_n)

Source: 32-th Vietnamese Mathematical Olympiad 1994

2/24/2007
Define the sequence {xn}\{x_{n}\} by x0=a∈(0,1)x_{0}=a\in (0,1) and xn+1=4π2(cos⁡−1xn+π2)sin⁡−1xn(n=0,1,2,...)x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...). Show that the sequence converges and find its limit.
trigonometrycalculuscalculus computations
Do there exist polynomials ?

Source: 32-th Vietnamese Mathematical Olympiad 1994

2/24/2007
Do there exist polynomials p(x),q(x),r(x)p(x), q(x), r(x) whose coefficients are positive integers such that p(x)=(x2−3x+3)q(x)p(x) = (x^{2}-3x+3) q(x) and q(x)=(x220−x15+112)r(x)q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)?
algebrapolynomialalgebra unsolved