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Special Numbers

Source: Cono Sur Olympiad 2016, problem 1

August 18, 2017
number theoryPerfect Squarescono sur

Problem Statement

Let abcd‾\overline{abcd} be one of the 9999 numbers 0001,0002,0003,…,9998,99990001, 0002, 0003, \ldots, 9998, 9999. Let abcd‾\overline{abcd} be an special number if ab−cdab-cd and ab+cdab+cd are perfect squares, ab−cdab-cd divides ab+cdab+cd and also ab+cdab+cd divides abcdabcd. For example 2016 is special. Find all the abcd‾\overline{abcd} special numbers. Note: If abcd‾=0206\overline{abcd}=0206, then ab=02ab=02 and cd=06cd=06.
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