Problem

Source: Mexican Mathematical Olympiad 2016 Day 1 Question 3

Tags: inequalities, floor function, algebra, national olympiad



Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$