Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Find the answer

[44.2] Find all pairs of positive integers $\{a, b\}$ , such that $a^{2} /\left(2 a b^{2}-b^{2}+1\right)$ is a positive integer.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

None

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.