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decimal and floor function ineq

Source: 2023 Mathematics Regional Olympiad of Mexico West P3

October 21, 2024

algebrafloor functioninequalities

Problem Statement

Let

x>1

be a real number that is not an integer. Denote

\{x\}

as its decimal part and

\lfloor x\rfloor

the floor function. Prove that

\left(\frac{x+\{x\}}{\lfloor x\rfloor}-\frac{\lfloor x\rfloor}{x+\{x\}}\right)+\left(\frac{x+\lfloor x\rfloor}{\{x\}}-\frac{\{x\}}{x+\lfloor x\rfloor}\right)>\frac{16}{3}