5. Two natural numbers $x$ and $y$ sum to 111, such that the equation
$$
\sqrt{x} \cos \frac{\pi y}{2 x}+\sqrt{y} \sin \frac{\pi x}{2 y}=0
$$

holds. Then a pair of natural numbers $(x, y)$ that satisfies the condition is
$\qquad$
5. $(37,74)$.

To make the equation hold, we need to consider the values of trigonometric functions at special angles.
Observe $\cos \frac{\pi y}{2 x}$.
When $y=2 x$, $\cos \frac{\pi y}{2 x}=\cos \pi=-1$. Let's experiment with the original equation by setting $y=2 x$.
The left side $=\sqrt{x} \cos \frac{2 x \pi}{2 x}+\sqrt{2 x} \sin \frac{x \pi}{2 \times 2 x}$
$=\sqrt{x} \cos \pi+\sqrt{2 x} \sin \frac{\pi}{4}$
$=-\sqrt{x}+\sqrt{2 x} \times \frac{\sqrt{2}}{2}=0=$ right side.
This exactly satisfies the requirement $\sqrt{x} \cos \frac{\pi y}{2 x}+\sqrt{y} \sin \frac{\pi x}{2 y}=0$.
Let $x+y=111$. Thus, $x=37, y=74$.
Therefore, a pair of natural numbers $(x, y)$ that meet the conditions of the problem is $(37,74)$.