My work: It's simple to see that, with the definition given, two numbers are romantic if $s(x) + s(y) = x + y$, where $s$ is the function that sums the digits of the number (e.g $s(13) = 1 + 3 = 4$). Now, if $ 10 \leq x, y \leq 99$, then $s(x) + s(y) \leq 18 + 18 = 36$. So we only need to look for solutions in the set $A = \{10, 12, 14, 16, \cdots, 26 \}$. For any $x, y \in A$, we have $20 \leq x + y \leq 52$, and so $2 \leq s(x) + s(y) \leq 18$. Therefore there are no $x, y \in A$ such that $x + y = s(x) + s(y)$. We conclude that there are no even romantic numbers.
Is this correct?