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4 problems match “triangle inequality”. Clear

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1. Minimum distance between points on circles ●○○○○ 10 ptsLet $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 5$. Find the smallest possible value of $|w - z|$.
2. Smallest modulus on a segment ●●●○○ 50 ptsLet $z$ be a complex number such that $|z - 12| + |z - 5i| = 13$. Find the smallest possible value of $|z|$. Round to two decimal places.
3. Minimum of a sum of |1+z^k| ●●●●○ 100 ptsLet the complex number $z$ satisfy $|z|=1$. Find the minimum possible value of $2021|1+z|+|1+z^{2}|+|1+z^{3}|+\cdots+|1+z^{4043}|$.
4. Minimum of a hyperbola distance expression ●●●●○ 100 ptsLet complex numbers $z_1$ and $z_2$ satisfy $\operatorname{Re} z_1>0$, $\operatorname{Re} z_2>0$, and $\operatorname{Re}(z_1^2)=\operatorname{Re}(z_2^2)=2$. Find the minimum value of $|z_1+2|+|\overline{z_2}+2|-|\overline{z_1}-z_2|$. Round to two decimal places.