A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two perpendicular chords with red endpoints.
2026-03-28 07:35:00.1774683300
Circle with perpendicular chords
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I think that this problem needs another assumption and that is it: the circumference is an even number.
We know that the proportion of sizes of two angels in a circle is equal to proportion of lengths of the parts that they dismember from the circle look at the image for better understanding!
so what we need to prove is "there are for different points that sum of length of the parts that they dismember from circle is half of its circumference" and that is not hard at all. I hop it helps:)