MathDB
Equidistributed sequence

Source: SEEMOUS 2019, problem 1

March 18, 2019
real analysiscollege contests

Problem Statement

A sequence {xn}n=1∞,0≤xn≤1\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1 is called "Devin" if for any f∈C[0,1]f\in C[0,1] lim⁡n→∞1n∑i=1nf(xi)=∫01f(x) dx \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx Prove that a sequence {xn}n=1∞,0≤xn≤1\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1 is "Devin" if and only if for any non-negative integer kk it holds lim⁡n→∞1n∑i=1nxik=1k+1.\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.
Remark. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.
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