2013 Gauss 8 Problem 24

In any triangle, the length of the longest side is less than half of the perimeter. All triangles with perimeter 57 and integer side lengths x, y, z, such that x < y < z are constructed. How many such triangles are there?

\textbf{(A)}\ 68\quad \textbf{(B)}\ 61\quad \textbf{(C)}\ 75\quad \textbf{(D)}\ 56\quad \textbf{(E)}\ 27

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Full credit to this problem is given to the CEMC, you may view all Gauss contests here.