Dvoryaninov S.

On a circle, 100 points are marked. Can it happen that there are exactly 1000 right triangles, all vertices of which are the marked points?

#
A triangle is right-angled if and only if some two of its vertices are diametrically opposite points on its circumscribed circle. Each pair of diametrically opposite points corresponds to exactly 98 right-angled triangles, and for different pairs, they are distinct. Therefore, the total number of right-angled triangles is divisible by 98, but 1000 is not divisible by 98.

## Answer

it cannot.