2024 AIME II Problem 14

Let b \ge 2 be an integer. Call a positive integer n b-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 \ge 2 for which there are more than ten b-eautiful integers.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.