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AIME 2024 / AIME 2024 train ad65db4f-5a65-56c6-b7c0-833a54140a20

Problem

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Problem

Let b≥2b \geq 2 be an integer. Call a positive integer nn bb\textit{-eautiful} if it has exactly two digits when expressed in base bb, and these two digits sum to n\sqrt{n}. For example, 8181 is 1313-eautiful because $81=\underline{6}\underline{3}_{13}$ and 6+3=816+3=\sqrt{81}. Find the least integer b≥2b \geq 2 for which there are more than ten bb-eautiful integers.
Plain-text mathematical notation (without MathML)
Let b≥2 be an integer. Call a positive integer n b\textit{-eautiful} if it has exactly two digits when expressed in base b, and these two digits sum to √(n). For example, 81 is 13-eautiful because $81=\underline{6}\underline{3}_{13}$ and 6+3=√(81). Find the least integer b≥2 for which there are more than ten b-eautiful integers.
Original LaTeX notation
Let $b \geq 2$ be an integer. Call a positive integer $n$ $b$\textit{-eautiful} if it has exactly two digits when expressed in base $b$, and these two digits sum to $\sqrt{n}$. For example, $81$ is $13$-eautiful because $81=\underline{6}\underline{3}_{13}$ and $6+3=\sqrt{81}$. Find the least integer $b \geq 2$ for which there are more than ten $b$-eautiful integers.

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Source and history

task_id
ad65db4f-5a65-56c6-b7c0-833a54140a20
task_key
default--train--ad65db4f-5a65-56c6-b7c0-833a54140a20
task_revision_id
3
config / split
default / train

Official source

initial import