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Print55rd Ukrainian National Mathematical Olympiad - Third Round
Ukraine algebra
Problem
Is it possible to compose two integer numbers using each of the ten digits exactly once, such that one of them is the square of the other?
0 cannot be the first digit in either number.
0 cannot be the first digit in either number.
Solution
If a number has digits, its square can contain no more than digits, which gives in total. If our number has digits or more, its square has at least , therefore, we must use at least digits. Therefore, no such number exists.
Final answer
No
Techniques
IntegersOther