MathDB
Romania District Olympiad 2011 - Grade XI

Source:

March 12, 2011
functionlimitcontinued fractionreal analysisreal analysis unsolved

Problem Statement

a) Prove that {x+y}−{y}\{x+y\}-\{y\} can only be equal to {x}\{x\} or {x}−1\{x\}-1 for any x,y∈Rx,y\in \mathbb{R}.
b) Let α∈R\Q\alpha\in \mathbb{R}\backslash \mathbb{Q}. Denote an={nα}a_n=\{n\alpha\} for all n∈N∗n\in \mathbb{N}^* and define the sequence (xn)n≥1(x_n)_{n\ge 1} by
xn=(a2−a1)(a3−a2)⋅…⋅(an+1−an)x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)
Prove that the sequence (xn)n≥1(x_n)_{n\ge 1} is convergent and find it's limit.
Back to ProblemsView on AoPS