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Repetitive Squares?

Can someone propose a solution to the following question using only 'basic' secondary school mathematics?

A repetitive number is a natural number that consists of two equal strings of digits 'glued' together. For example, 99 and 998998 are repetitive numbers, but 99099 is not. Are there any repetitive perfect squares? If so, how many?

Note by H K
5 months ago

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See 1988 IMO Shortlist, #25. Jon Haussmann · 5 months ago

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@Jon Haussmann Indeed. This is a problem that once you write down (in a mathematical equation) what you need, it essentially resolves itself.

To get you started:
Let \( 10 ^k \leq a < 10^{k+1} \).
We want to know if \( ( (10 ^{k+1} + 1 ) \times a \) could ever be a square. Calvin Lin Staff · 5 months ago

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