This is a question from Bosnia and Hercegovina mathematical Olympiad $2002$.
There are already many similar questions, which I have done in the past.
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.