The number of isosceles triangles with integer sides if no side exceeds 2008 is

If the sides are a,a,b then the triangle forms only

when 2a > b. So for any$a\in N,b$ can change from 1 to 2a –1, where $a\le 1004\Rightarrow \text{ no. }$of triangles = 1 + 2 +
3 + — +(2(1004)–1) =$(1004{)}^2$ 

And if $1005\le a\le 2008,b$ can take any value from 1 to 2008

But a has 1004 possibilities hence

No. of triangles$=1004\times 2008=2(1004{)}^2$

$\therefore$ Total no. of isosceles triangles $=3(1004{)}^2$