The sixty-seventh question: Given a positive integer $n$, let $z_{1}, z_{2}, \ldots, z_{n}$ be $n$ complex numbers with modulus not greater than 1. Try to find the maximum possible value of $\left|\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{z}_{\mathrm{k}}^{2}\right|-\left|\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{z}_{\mathrm{k}}\right|^{2}$.
The 67th problem,
Solution: We first prove a lemma: If real numbers $x_{k} \in[-1,1](1 \leq k \leq n)$, then the maximum value of $\sum_{k=1}^{n} x_{k}^{2}-\left(\sum_{k=1}^{n} x_{k}\right)^{2}$ is $2\left[\frac{n}{2}\right]$.

Proof of the lemma: For any $i \in\{1,2, \ldots, n\}$, consider $x_{i}$ as a variable and the rest $x_{j}(1 \leq j \leq n, j \neq i)$ as constants, then
$$
\sum_{k=1}^{n} x_{k}^{2}-\left(\sum_{k=1}^{n} x_{k}\right)^{2}=-2 \sum_{1 \leq i<j \leq n} x_{i} x_{j}
$$

is a linear function of $x_{i}$, and it clearly achieves its maximum value when $x_{i} \in\{-1,1\}$. Therefore, we only need to consider the case where each $x_{i}$ $(1 \leq i \leq n)$ is either 1 or -1.
Suppose $x_{1}, x_{2}, \ldots, x_{n}$ have exactly $s$ ones and $t$ negative ones, then
$$
\sum_{k=1}^{n} x_{k}^{2}-\left(\sum_{k=1}^{n} x_{k}\right)^{2}=n-(s-t)^{2} \leq n-\left(n-2\left[\frac{n}{2}\right]\right)=2\left[\frac{n}{2}\right]
$$

Equality holds when $x_{1}, x_{2}, \ldots, x_{n}$ have exactly $\left[\frac{n}{2}\right]$ ones and $\left\lceil\frac{n}{2}\right]$ negative ones.
Now, let's return to the original problem.
Rotate the coordinate system so that $\sum_{k=1}^{n} z_{k}^{2} \in \mathbb{R}$ and $\sum_{k=1}^{n} z_{k}^{2} \geq 0$. Let $z_{k}=x_{k}+iy_{k}(1 \leq k \leq n)$, then
$$
\sum_{k=1}^{n} z_{k}^{2}=\sum_{k=1}^{n}\left(x_{k}+i y_{k}\right)^{2}=\sum_{k=1}^{n}\left(x_{k}^{2}-y_{k}^{2}\right)+2\left(\sum_{k=1}^{n} x_{k} y_{k}\right) i=\sum_{k=1}^{n}\left(x_{k}^{2}-y_{k}^{2}\right)
$$

We also know:
$$
\begin{array}{l}
\left|\sum_{k=1}^{n} z_{k}\right|^{2}=\left|\left(\sum_{k=1}^{n} x_{k}\right)+i\left(\sum_{k=1}^{n} y_{k}\right)\right|^{2} \\
=\left(\sum_{k=1}^{n} x_{k}\right)^{2}+\left(\sum_{k=1}^{n} y_{k}\right)^{2}
\end{array}
$$

Thus, according to the lemma,
$$
\begin{array}{l}
\left|\sum_{k=1}^{n} z_{k}^{2}\right|-\left|\sum_{k=1}^{n} z_{k}\right|^{2}=\sum_{k=1}^{n}\left(x_{k}^{2}-y_{k}^{2}\right)-\left(\left(\sum_{k=1}^{n} x_{k}\right)^{2}+\left(\sum_{k=1}^{n} y_{k}\right)^{2}\right) \\
\leq \sum_{k=1}^{n} x_{k}^{2}-\left(\sum_{k=1}^{n} x_{k}\right)^{2} \leq\left\lceil\frac{n}{2}\right]
\end{array}
$$

Equality holds when $z_{1}, z_{2}, \ldots, z_{n}$ have exactly $\left[\frac{n}{2}\right]$ ones and $\left\lceil\frac{n}{2}\right]$ negative ones.