MathDB

P6

Part of 2021 Thailand Online MO

Problems(1)

Sequence of rational numbers cannot be too close

Source: 2021 Thailand Online MO P6 (Mock TMO contest)

4/6/2021
Let m<nm<n be two positive integers and xm<xm+1<⋯<xnx_m<x_{m+1}<\cdots<x_n be a sequence of rational numbers. Suppose that kxkkx_k is an integer for all integers kk which m≤k≤nm\leq k\leq n. Prove that xn−xm≥1m−1n.x_n-x_m\geq \frac{1}{m}-\frac{1}{n}.
algebra