If the perimeter of an isosceles triangle with integral sides is 2017, how many different lengths are possible for the legs

Because the perimeter is 2017, the side lengths must add to 2017;
a+b+c=2017
An isosceles triangle has two sides of the same length, so b=a. These two sides of the same length are the "legs".
a+a+c=2017
2a+c=2017
Integral means integer (assuming this is not a calculus problem). So, the smallest possible length for side c is 1.
2a+1=2017
2a=2016
a=1008
The legs can be up to 1008 units long apiece. Because the legs must be integer values, 1008 different lengths are possible for the legs.