Find the number of isosceles triangles with integral sides(each side ≤2010)

i think that you have not specified that which side of the triangle is less than or equal to 2010 but if equal sides are taken as less than or equal to 2010 then number of isosceles triangle that can be formed are 4019 because in a triangle sum of two sides is always greater than third side ,so third side can be 1,2,3,4,5.....................till 4019.and there are 4019 integral numbers.

Hi Halak,

The base (ie is the un-equal side of the triangle could be 1,2,3,....,2010).

When base = 1:

No of triangle is 2010. [As the equal side can take values 1,2,3,.....,2010)

When base = 2:

No of triangles = 2009 [As the equal side can take values 2,3,4,....,2010).

When base = 3:

No of trianles = 2008 [As the equal side can take values 3,4,5,......2010).

And so on.....

So total number of triangles will be 2010+2009+2008+2007+2006+.....+2+1 = (2010*2011)/2 = 2011*1005 = 2021055.

Hence there can be 2021055 isoceles triangles with the given condition.

Hope this helps.

Regards,

Ashwin (IIT Madras).