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Part of 2022 Kosovo & Albania Mathematical Olympiad

Problems(1)

Positive integer solutions to inequality

Source: KAMO 2022 Grade 9 (answer-only question)

7/5/2022
Let a>0a>0. If the inequality 22<ax<22222<ax<222 holds for precisely 1010 positive integers xx, find how many positive integers satisfy the inequality 222<ax<2022222<ax<2022?
Note: The first 8 problems of the competition are questions which the contestants are expected to solve quickly and only write the answer of. This problem turned out to be a lot more difficult than anticipated for an answer-only question.
inequalities