  An old conjecture of Bollob\'as and Scott asserts that every Eulerian
directed graph with average degree $d$ contains a directed cycle of length at
least $\Omega(d)$. The best known lower bound for this problem is
$\Omega(d^{1/2})$ by Huang, Ma, Shapira, Sudakov and Yuster. They asked whether
this estimate can be improved at least for directed paths instead of cycles and
whether one can find a long path starting from any vertex if the host digraph
is connected. In this paper we break the $\sqrt{d}$ barrier, showing how to
find a path of length $\Omega(d^{1/2+1/40})$ from any vertex of a connected
Eulerian digraph.
