Problem 1306

1. Start with the given equation:
$$\sqrt{x^2 + 5} - x = \frac{1}{3}$$

2. Rearrange the equation to isolate the square root term:
$$\sqrt{x^2 + 5} = x + \frac{1}{3}$$

3. Square both sides to eliminate the square root:
$$(\sqrt{x^2 + 5})^2 = \left(x + \frac{1}{3}\right)^2$$
$$x^2 + 5 = x^2 + \frac{2}{3}x + \frac{1}{9}$$

4. Simplify the equation by subtracting $x^2$ from both sides:
$$5 = \frac{2}{3}x + \frac{1}{9}$$

5. Clear the fraction by multiplying every term by 9:
$$45 = 6x + 1$$

6. Isolate $x$ by subtracting 1 from both sides:
$$44 = 6x$$

7. Solve for $x$ by dividing both sides by 6:
$$x = \frac{44}{6} = \frac{22}{3}$$

8. Verify that $x = \frac{22}{3}$ satisfies the original equation:
$$\sqrt{\left(\frac{22}{3}\right)^2 + 5} - \frac{22}{3} = \frac{1}{3}$$
$$\sqrt{\frac{484}{9} + 5} - \frac{22}{3} = \frac{1}{3}$$
$$\sqrt{\frac{484}{9} + \frac{45}{9}} - \frac{22}{3} = \frac{1}{3}$$
$$\sqrt{\frac{529}{9}} - \frac{22}{3} = \frac{1}{3}$$
$$\frac{23}{3} - \frac{22}{3} = \frac{1}{3}$$
$$\frac{1}{3} = \frac{1}{3}$$

The solution is verified.

9. The fraction $x = \frac{22}{3}$ has a numerator of 22 and a denominator of 3. The sum of the numerator and denominator is:
$$22 + 3 = 25$$

The final answer is $\boxed{25}$.