Problem
COM-B2-M07-P016 Euler Circuit: Sufficiency
Prove that if a finite connected graph has only even degrees, then it has a closed walk using every edge exactly once.
Take a maximal closed walk and, if edges remain, splice in another cycle.
Start at any vertex and follow unused edges as long as possible. Since all degrees are even, one cannot get stuck at a vertex different from the start: every entrance requires another exit. Thus we obtain a closed walk.
Choose a closed walk using as many edges as possible. If it does not use all edges, connectedness gives a vertex of the walk incident to an unused edge leading into the unused part. Starting there and moving along unused edges, we again obtain a closed walk, which can be spliced into the first one. This increases the number of used edges, a contradiction.
Therefore the maximal walk uses all edges.
This is a full olympiad lemma, not just a fact.