SEEMOUS 2019 Problem 1

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