Ten circles are all the same size. Each pair of these circles overlap but no circle is exactly on top of another circle. What is the greatest possible total number of intersection points of these ten circles?
Each pair intersect in 2 points.
The first circle intersects 9 others.
The 2nd circle intersects 8 others (not counting its intersections with the first circle)
The 3rd circle intersects 7 others (not counting its intersections with the first two)
The 9th circle intersects 1 other (not counting ...etc)
The 10th circle already intersects all the others.
Hence number of intersection points = 2*(9+8+7+6+5+4+3+2+1) = 2*45 = 90.