Problem: On a circle, Alina draws $2019$ chords, the endpoints of which are all different.
A point is considered marked if it is either
\begin{enumerate}[(i)]
  \ii one of the $4038$ endpoints of a chord; or
  \ii an intersection point of at least two chords.
\end{enumerate}
Of the $4038$ points meeting criterion (i),
Alina labels $2019$ points with a $0$ and the other $2019$ points with a $1$.
She labels each point meeting criterion (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 $0, 1, \dots, N$ exactly once.
Show that at least one blue label is a multiple of $3$.

\end{enumerate}

Solution: Only the labels mod $3$ matter at all.

Assume for contradiction no blue labels are divisible by $3$.
Let $e_{ij}$ denote the number of segments
joining $i \pmod 3$ to $j \pmod 3$.
By double-counting (noting that points in (ii) are
counted an even number of times
but points in (i) are counted once)
we derive that
\begin{align*}
  e_{01} + e_{02} &\equiv 2019 \pmod 2 \\
  e_{01} + e_{12} &\equiv 2019 \pmod 2 \\
  e_{02} + e_{12} &\equiv 0 \pmod 2
\end{align*}
which gives \[ e_{02} \equiv e_{12} \equiv 1 - e_{01} \pmod 2. \]
However, one can check this is incompatible
with the hypothesis that the yellow labels
are $0$, $1$, \dots, $N$.

\begin{remark*}
  In addition, we can replace the points/segments
  by any graph $G$ for which there are
  \begin{itemize}
  \ii an odd number of leaves (or just odd-degree vertices) labeled $0$,
  \ii an odd number of leaves (or just odd-degree vertices) labeled $1$,
  \ii and the remaining vertices have even degree.
  \end{itemize}
  Thus the geometry of the problem is smoke and mirrors, too.
\end{remark*}