a^2+b^2+c^2=121 and the question is: what is a+b+c=?

The answe is 17 but I don´t know how to prove it and which sort of method I should use. It should be quite easy since the course as a whole is considered to be basic

Yes, I totally get that, but then is when you are aware of the answer, which is not the case from the beginning. So, how do I prove that without being aware of the asnwer?

Without giving the condition that a+b+c=17, I don't think that you can formally "prove" it since there are other solutions that will satisfy the a^2+b^2+c^2=121. Other solutions could be a=0, b=0, c=11 and a=6, b=6 and c=7.....and so on.