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na-a is positive integer, period of 10 digits

Source: Argentina 2006 OMA L3 p1

May 12, 2024
algebranumber theoryperiodic

Problem Statement

Let AA be the set of positive real numbers less than 11 that have a periodic decimal expansion with a period of ten different digits. Find a positive integer nn greater than 11 and less than 101010^{10} such that naāˆ’ana-a is a positive integer for all aa. of set AA.
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