MathDB

3

Part of 1993 Vietnam National Olympiad

Problems(2)

find a function

Source: 31-th Vietnamese Mathematical Olympiad 1993

2/17/2007
Find a function f(n)f(n) on the positive integers with positive integer values such that f(f(n))=1993n1945f( f(n) ) = 1993 n^{1945} for all nn.
functionnumber theory unsolvednumber theory
two sequence

Source: 31-th Vietnamese Mathematical Olympiad 1993

2/17/2007
Define the sequences a0,a1,a2,...a_{0}, a_{1}, a_{2}, ... and b0,b1,b2,...b_{0}, b_{1}, b_{2}, ... by a0=2,b0=1,an+1=2anbn/(an+bn),bn+1=an+1bna_{0}= 2, b_{0}= 1, a_{n+1}= 2a_{n}b_{n}/(a_{n}+b_{n}), b_{n+1}= \sqrt{a_{n+1}b_{n}}. Show that the two sequences converge to the same limit, and find the limit.
invarianttrigonometryreal analysisreal analysis unsolved