Olympiad Maths Prep

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Geometry Difficulty 5.7 AIME, harder Prove it Ukraine

11 points are given on a circle. Petrik numbered them by the numbers 1,2,…,111, 2, \ldots, 11. After that, pairs of points were connected by segments: 11 and 22, 22 and 33, …\ldots, 1010 and 1111, 1111 and 11. What is the largest possible number of intersection points of these segments? The given 11 points are not counted as intersection points.

Figure 1
Fig. 4

Solution

Consider one of the drawn segments. It cannot intersect itself, nor the adjacent segments on either side, e.g., the segment 33–44 does not intersect the segments 22–33, 33–44, and 44–55. Thus the maximum number of possible intersection points occurs when each of the segments intersects all the other 88 non-adjacent segments. This can be achieved by numbering the points around the circle in the following way (see Fig. 4):
1−3−5−7−9−11−2−4−6−8−10−1. 1 - 3 - 5 - 7 - 9 - 11 - 2 - 4 - 6 - 8 - 10 - 1.
The number of intersection points is equal to 12⋅11⋅8=44\frac{1}{2} \cdot 11 \cdot 8 = 44.

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