Product of $10$ consecutive integers can never be a perfect square

Question

This is a question from Bosnia and Herzegovina mathematical Olympiad $2002$.

There are already many similar questions, which I have done in the past.

For $5$ consecutive integers

A more general case for $3$ consecutive integers.

For $6$ consecutive integers

And similar proves are available for $3$ (Which was once asked in Korean Mathematical Olympiad) and $4$ (Indian national mathematical Olympiad) consecutive integers.

I also know the general result which is here but I am no way going to use that in a competition.

Please try to give a high school argument as supposed to be given in a national Olympiad.

Thanks.

Answer 1

Ah got it. Here is a Wordpress article which gives a proof which is not so simple but works fine at Olympiad level.