ELMO 2017 Problem 5

The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels.

(Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.)

Proposed by Michael Ma