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Obtain the AM of the digits by putting a decimal separator

Source: MEMO 2022 T8

September 2, 2022
number theory

Problem Statement

We call a positive integer <spanclass=′latex−italic′>cheesy</span><span class='latex-italic'>cheesy</span> if we can obtain the average of the digits in its decimal representation by putting a decimal separator after the leftmost digit. Prove that there are only finitely many <spanclass=′latex−italic′>cheesy</span><span class='latex-italic'>cheesy</span> numbers.
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