MathDB
CIIM 2019 Problem 3

Source:

August 30, 2021
CIIM

Problem Statement

Let {an}n∈N\{a_n\}_{n\in \mathbb{N}} a sequence of non zero real numbers. For m≥1m \geq 1, we define: Xm={X⊆{0,1,…,m−1}:∣∑x∈Xax∣>1m}. X_m = \left\{X \subseteq \{0, 1,\dots, m - 1\}: \left|\sum_{x\in X} a_x \right| > \dfrac{1}{m}\right\}. Show that lim⁡n→∞∣Xn∣2n=1.\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.
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