Problem

Source: 2024 Vietnam National Olympiad - Problem 1

Tags: algebra, polynomial



For each real number $x$, let $\lfloor x \rfloor$ denote the largest integer not exceeding $x$. A sequence $\{a_n \}_{n=1}^{\infty}$ is defined by $a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.$ Let $b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.$ a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1$. b) Prove that there exists a strictly increasing sequence $\{n_k \}_{k=1}^{\infty}$ of positive integers such that $$\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.$$