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Difference between revisions of "1970 IMO Problems/Problem 2"

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[[Category:Olympiad Number Theory Problems]]
 
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[[Category:Olympiad Inequality Problems]]

Revision as of 14:56, 29 October 2007

Problem

Let $a, b$, and $n$ be integers greater than 1, and let $a$ and $b$ be the bases of two number systems. $A_{n-1}$ and $A_{n}$ are numbers in the system with base $a$ and $B_{n-1}$ and $B_{n}$ are numbers in the system with base $b$; these are related as follows:

$A_{n} = x_{n}x_{n-1}\cdots x_{0}, A_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}$,

$B_{n} = x_{n}x_{n-1}\cdots x_{0}, B_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}$,

$x_{n} \neq 0, x_{n-1} \neq 0$.

Prove:

$\frac{A_{n-1}}{A_{n}} < \frac{B_{n-1}}{B_{n}}$ if and only if $a > b$.

Solution

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1970 IMO (Problems) • Resources
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