International Mathematical Olympiad 1970 Problem 2

DigitsInequalitiesNT

Let a,ba, b and nn be integers greater than 1, and let aa and bb be the bases of two number systems. An−1A_{n-1} and AnA_n are numbers in the system with base aa, and Bn−1B_{n-1} and BnB_n are numbers in the system with base bb; these are related as follows: An=xnxn−1⋯x0,An−1=xn−1xn−2⋯x0,A_n = x_n x_{n-1} \cdots x_0, \quad A_{n-1} = x_{n-1} x_{n-2} \cdots x_0, Bn=xnxn−1⋯x0,Bn−1=xn−1xn−2⋯x0,B_n = x_n x_{n-1} \cdots x_0, \quad B_{n-1} = x_{n-1} x_{n-2} \cdots x_0, xn≠0,xn−1≠0.x_n \neq 0, \quad x_{n-1} \neq 0.

Prove: An−1An<Bn−1Bn if and only if a>b.\frac{A_{n-1}}{A_n} < \frac{B_{n-1}}{B_n} \text{ if and only if } a > b.