Problem

Source: 2025 Vietnam National Olympiad - Problem 1

Tags: algebra, polynomial, Sequence



Let $P(x) = x^4-x^3+x$. a) Prove that for all positive real numbers $a$, the polynomial $P(x) - a$ has a unique positive zero. b) A sequence $(a_n)$ is defined by $a_1 = \dfrac{1}{3}$ and for all $n \geq 1$, $a_{n+1}$ is the positive zero of the polynomial $P(x) - a_n$. Prove that the sequence $(a_n)$ converges, and find the limit of the sequence.