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3

Part of 2010 IMC

Problems(2)

IMC 2010 - Problem 3

Source:

7/26/2010
Define the sequence x1,x2,...x_1, x_2, ... inductively by x1=5x_1 = \sqrt{5} and xn+1=xn2−2x_{n+1} = x_n^2 - 2 for each n≥1n \geq 1. Compute lim⁡n→∞x1⋅x2⋅x3⋅...⋅xnxn+1\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}.
limitfunctionreal analysisreal analysis unsolved
IMC 2010 Problem 3, Day 2

Source:

7/27/2010
Denote by SnS_n the group of permutations of the sequence (1,2,…,n).(1,2,\dots,n). Suppose that GG is a subgroup of Sn,S_n, such that for every π∈G∖{e}\pi\in G\setminus\{e\} there exists a unique k∈{1,2,…,n}k\in \{1,2,\dots,n\} for which π(k)=k.\pi(k)=k. (Here ee is the unit element of the group Sn.S_n.) Show that this kk is the same for all π∈G∖{e}.\pi \in G\setminus \{e\}.
group theoryabstract algebrainductionstrong inductionsuperior algebrasuperior algebra unsolved