Is it possible to select 100 points on a circle so that there are exactly 1000 right triangles with the vertices at selected points?
Sergey Dvoryaninov
The only time three points on a circle form a right-angled triangle is when two of them form a diameter (which is the hypotenuse). Since there are $100$ points, one diameter will form $98$ right angled triangles. Hence the number of triangles formed from $n$ diameters is $98n$. However $1000$ is not divisible by $98$, so it is impossible to form exactly $1000$ right angled triangles with $100$ points.