MathDB

On a circle, Alina draws

2019

chords, the endpoints of which are all different. A point is considered marked if it is either

\text{(i)}

one of the

4038

endpoints of a chord; or

\text{(ii)}

an intersection point of at least two chords.
Alina labels each marked point. Of the

4038

points meeting criterion

\text{(i)}

, Alina labels

2019

points with a

0

and the other

2019

points with a

1

. She labels each point meeting criterion

\text{(ii)}

with an arbitrary integer (not necessarily positive). Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with

k

marked points has

k-1

such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference. Alina finds that the

N + 1

yellow labels take each value

0, 1, . . . , N

exactly once. Show that at least one blue label is a multiple of

3

. (A chord is a line segment joining two different points on a circle.)

Problem Statement