Olympiad Maths Prep

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Algebra Difficulty 5.3 AIME, harder Prove it Ukraine

Is it possible to compose two integer numbers using each of the ten digits 0,1,…,90, 1, \ldots, 9 exactly once, such that one of them is the square of the other?

*0 cannot be the first digit in either number.*

Solution

If a number has 33 digits, its square can contain no more than 66 digits, which gives 99 in total. If our number has 44 digits or more, its square has at least 77, therefore, we must use at least 1111 digits. Therefore, no such number exists.

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