1. **Problem statement:** Given complex numbers $z_1, z_2, z_3$ with magnitudes $|z_1|=1$, $|z_2|=2$, $|z_3|=3$ and sum $z_1 + z_2 + z_3 = 3 + \sqrt{5}i$, find the value of $\operatorname{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1})$. 2. **Key formulas and rules:** - For any complex number $z = x + yi$, $|z|^2 = z \overline{z} = x^2 + y^2$. - The real part of a complex number $w$ is $\operatorname{Re}(w) = \frac{w + \overline{w}}{2}$. - The magnitude squared of the sum is $|z_1 + z_2 + z_3|^2 = (z_1 + z_2 + z_3)(\overline{z_1} + \overline{z_2} + \overline{z_3})$. 3. **Calculate $|z_1 + z_2 + z_3|^2$:** $$ |3 + \sqrt{5}i|^2 = 3^2 + (\sqrt{5})^2 = 9 + 5 = 14 $$ 4. **Expand $|z_1 + z_2 + z_3|^2$ using the sum:** $$ |z_1 + z_2 + z_3|^2 = (z_1 + z_2 + z_3)(\overline{z_1} + \overline{z_2} + \overline{z_3}) $$ Expanding: $$ = |z_1|^2 + |z_2|^2 + |z_3|^2 + z_1 \overline{z_2} + z_2 \overline{z_1} + z_2 \overline{z_3} + z_3 \overline{z_2} + z_3 \overline{z_1} + z_1 \overline{z_3} $$ 5. **Substitute known magnitudes:** $$ = 1^2 + 2^2 + 3^2 + (z_1 \overline{z_2} + z_2 \overline{z_1} + z_2 \overline{z_3} + z_3 \overline{z_2} + z_3 \overline{z_1} + z_1 \overline{z_3}) = 1 + 4 + 9 + \text{sum of cross terms} = 14 + \text{sum of cross terms} $$ 6. **Note that the sum of cross terms is twice the real part of $z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}$:** $$ z_1 \overline{z_2} + z_2 \overline{z_1} + z_2 \overline{z_3} + z_3 \overline{z_2} + z_3 \overline{z_1} + z_1 \overline{z_3} = 2 \operatorname{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) $$ 7. **Set up the equation:** $$ |z_1 + z_2 + z_3|^2 = 14 = 14 + 2 \operatorname{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) $$ 8. **Solve for the real part:** $$ 14 = 14 + 2 \operatorname{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \\ 0 = 2 \operatorname{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \\ \operatorname{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 0 $$ **Final answer:** $$\boxed{0}$$