Question

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

Answer 1

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.