Several chords are drawn in a circle of radius 1 such that each diameter intersects no more than 4 of them. Prove that the sum of their lengths does not exceed 13.
2026-04-01 16:22:04.1775060524
Pigeonhole principle problem involving circle and its chords
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1
Let us suppose that chords are drawn in a circle of radius 1 such that each diameter intersects no more than 4 of them and the sum of their lengths exceeds 13. (it will be shown that this leads to contradiction)
For each chord draw a smaller circular arc that connects end points of the chord, and also a circular arc that is symmetric to the former about the center of the circle. This is illustrated on following picture:
Here we can notice that any diameter that belongs to ABB'A' crosses the chord in question (AB).
Also, length of the circular arcs is in general greater that the length of the chord. In our case, this means that the sum of lengths of all such circular arcs for all chords is greater than 2 * 13 = 26.
Since the length of the circle is 2 * PI and 26 > 4 * (2 * PI), there must be a point on the circle that will be within more than 4 such circular arcs. According to initial method of choosing circular arcs, diameter that goes through that point will intersect more than 4 chords. Contradiction.
Hence, the statement of the problem must be true.
NOTE: I wanted to explore how we can create problems of type:
Several chords are drawn in a circle of radius 1 such that each diameter intersects no more than N of them. Prove that the sum of their lengths does not exceed L.
The key relation is 2 * L > N * (2 * PI), or L > PI * N. So another problems could be:
Several chords are drawn in a circle of radius 1 such that each diameter intersects no more than 3 of them. Prove that the sum of their lengths does not exceed 10.
Several chords are drawn in a circle of radius 1 such that each diameter intersects no more than 31 of them. Prove that the sum of their lengths does not exceed 100.
Several chords are drawn in a circle of radius 1 such that each diameter intersects no more than 641 of them. Prove that the sum of their lengths does not exceed 2014.