Chords of a 20-gon

Question

Twenty points lie on a circle, so as to form a regular polygon. Then they are split into ten pairs, and the points in each pair are connected by a chord. Prove that some pair of these chords have the same length.

At first i thought this was be a simple case of pigeon hole principle. But it is not that, as there are 10 possible lengths and there are 10 chords.

Answer 1

We label the distances $1,2,3\dots 10$ according to the congruence of $v_1-v_2\bmod10$. Suppose we have split it up. Then

$v_1-v_2+v_3-v_4+\dots v_{19}-v_{20}=v_1+v_2+v_3+\dots+v_{20}-2(v_2+v_4+v_6+\dots v_{20})\equiv 0\bmod 2$

However if we had distances $1,2,3\dots10$ the sum of the distances would be $55$ so it is impossible to do this.

Answer 2

If we're talking about the chords of the 20-gon (dodecagon), then the chords between ajacent points don't count (they don't lie inside the polygon) as chords and there are only 9 possible lengths.

If the chords are on the circle then it's definitely more complicated as you suggest.