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Find digits x,y,z such that equality holds

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September 27, 2010
number theorydecimal representationequationIMO ShortlistIMO Longlist

Problem Statement

Find digits x,y,zx, y, z such that the equality xx⋯x‾⏟2n times−yy⋯y‾⏟n times=zz⋯z‾⏟n times\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}} holds for at least two values of n∈Nn \in \mathbb N, and in that case find all nn for which this equality is true.
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