# How many isosceles triangles with total side length $100$ are there?

Let the sum of the three sides of a triangle be $100,$ and all the sides are positive integers length, how many possible isosceles triangles are there?

• Hint: count solutions to the (in)equations $2a+b=100$ and $2a \geq b$. – Daniel Franke Aug 3 '13 at 2:36
• @DanielFranke $2a>b$. – S.B. Aug 3 '13 at 2:37

Hint: We can certainly do $(49,49,2)$, $(48,48,4)$, $(47,47,6)$ and so on for a while. But the sum of the two equal sides cannot be less than or equal to the third side. That should tell you where we need to stop.
Hint: Let the side length $=x$, then we know the base length $=100-2x$. Then set up your inequality(s).
• How will the side lengths $1,1,98$ form a triangle? – ronno Aug 4 '13 at 10:42