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Ten points are chosen on a circle. A line segment is drawn between every pair of chosen points.

Every time two line segments intersect inside the circle, that intersection point is marked red. What is the largest number of points that can be marked red?


For example, for five points on the circle, we can get up to five red intersection points.

 

[asy] unitsize(2 cm);pair[] A;pair P;int i, j;A[1] = dir(18);A[2] = dir(60);A[3] = dir(124);A[4] = dir(190);A[5] = dir(330);A[6] = A[1];A[7] = A[2];A[8] = A[3];A[9] = A[4];draw(Circle((0,0),1));for (i = 1; i <= 5; ++i) { dot(A[i]);for (j = i + 1; j <= 5; ++j) { draw(A[i]--A[j]);} } for (i = 1; i <= 5; ++i) { P = extension(A[i], A[i + 2], A[i + 1], A[i + 3]);dot(P,red);} [/asy]

So how I did this, was that I figured out how many line segments there were (45), and after figuring that out, I did 45 choose 2, (cuz you need two line segments to intersect) and I got 990, which was wrong. I also tried 45^2, which was also incorrect.

 Jan 25, 2020
 #1
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I believe that these form the "pentalope numbers":  1, 5, 15, 35, 70, 126, 210, ...

 

Since there won't be any line segments that connect non-adjacent points until you have at least 4 points:

4 points will result in 1 point of intersection,

5 points will result in 5 points of intersection,

6 points will result in 15 points of intersection,

...

10 points will result in 210 points of intersection.

 

The formula is:  [n · (n + 1) · (n + 2) · (n + 3) ] / 24

 Jan 25, 2020

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