Given that a duck found that $5-2\sqrt{3}i$ is one of the roots of $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots?

[i]2022 CCA Math Bonanza Lightning Round 2.1[/i]
1. **Identify the polynomial and given root:**
   The polynomial is \( P(x) = x^3 - 17x^2 + 107x - 259 \). We are given that \( 5 - 2\sqrt{3}i \) is one of the roots.

2. **Use the fact that coefficients of polynomials with real coefficients are real:**
   Since the polynomial has real coefficients, the complex roots must occur in conjugate pairs. Therefore, if \( 5 - 2\sqrt{3}i \) is a root, then \( 5 + 2\sqrt{3}i \) must also be a root.

3. **Apply Vieta's formulas:**
   Vieta's formulas tell us that the sum of the roots of the polynomial \( x^3 - 17x^2 + 107x - 259 \) is equal to the coefficient of \( x^2 \) term (with a sign change), divided by the leading coefficient. For our polynomial, this is:
   \[
   \text{Sum of the roots} = -\frac{-17}{1} = 17.
   \]

4. **Sum of the known roots:**
   We know two of the roots: \( 5 - 2\sqrt{3}i \) and \( 5 + 2\sqrt{3}i \). The sum of these two roots is:
   \[
   (5 - 2\sqrt{3}i) + (5 + 2\sqrt{3}i) = 5 + 5 = 10.
   \]

5. **Determine the sum of the real parts of the other root:**
   Let the third root be \( r \). According to Vieta's formulas, the sum of all three roots is 17. Therefore, the sum of the real parts of the other root is:
   \[
   17 - 10 = 7.
   \]

6. **Conclusion:**
   The sum of the real parts of the other two roots is \( 7 \).

The final answer is \( \boxed{7} \).