## Triangles containing the origin

### Problem 184

Published on Friday, 29th February 2008, 09:00 pm; Solved by 1006; Difficulty rating: 75%Consider the set `I _{r}` of points (

`x`,

`y`) with integer co-ordinates in the interior of the circle with radius

`r`, centered at the origin, i.e.

`x`

^{2}+

`y`

^{2}<

`r`

^{2}.

For a radius of 2, `I`_{2} contains the nine points (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1) and (1,-1). There are eight triangles having all three vertices in `I`_{2} which contain the origin in the interior. Two of them are shown below, the others are obtained from these by rotation.

For a radius of 3, there are 360 triangles containing the origin in the interior and having all vertices in `I`_{3} and for `I`_{5} the number is 10600.

How many triangles are there containing the origin in the interior and having all three vertices in `I`_{105}?