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First, prove that if we have any set of 2n points on a circle, divided into two subsets A and B of n points each, then there is a way to pair each element of A with an element of B such that the chords between the paired elements do not intersect. It's easy by induction.
It's obviously true for n = 1.
Assume it's true for n and suppose that we have 2(n+1) points on the circle divided into two sets of size n+1 each. There clearly must be an element, a, of A which is adjacent to an element, b, of B.
We will pair a and b, and then use the induction hypothesis to pair up A-{a} and B-{b}.
None of the latter n chords intersect by the induction hypothesis, and none of them intersect the chord between a and b since those points are adjacent.
Now the problem's solution is immediate by letting A be the set of points numbered from 51 to 150 and B be all of the other points. |
