  Denote by $R(G_1, G_2, G_3)$ the minimum integer $N$ such that any
three-colouring of the edges of the complete graph on $N$ vertices contains a
monochromatic copy of a graph $G_i$ coloured with colour $i$ for some
$i\in{1,2,3}$. In a series of three papers of which this is the third, we
consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity.
Specifically, in this in this paper, we consider $R(C_n,C_m,C_{\ell})$, where
$n$ is even and $m$ and $\ell$ are odd. Figaj and \L uczak determined an
asymptotic result for this case, which we improve upon to give an exact result.
We prove that for $n,m$ and $\ell$ sufficiently large
$R(C_n,C_m,C_\ell)=\max\{4n-3, n+2m-3, n+2\ell-3\}$.
