let $a$ be legs, $b$ be the base.  $a: 1,2,3,4,5,6,7,8,9,10,11$  $b:21,19,17,15,13,11,9,7,5,3,1$  Since $2a\textgreater b$, $a\textgreater5$, there are $6$ possible $a$ 
How many different isosceles triangles have integer side lengths and perimeter $23$ ? (2005 AMC 8 Problem, Question \#15) 
Options:
[1] $$2$$ 
[2] $$4$$ 
[3] $$6$$ 
[4] $$9$$ 
[5] $$11$$ 