The 25th question: Let $\left(a_{1}, a_{2}, \ldots, a_{100}\right)$ be a permutation of $\{1,2, \ldots, 100\}$, and let $L(k)$ be the largest $m$ such that there exist $k=i_{1}<i_{2}<\ldots<i_{m}$ with $a_{i_{1}}>a_{i_{2}}>\ldots>a_{i_{m}}$ or $a_{i_{1}}<a_{i_{2}}<\ldots<a_{i_{m}}$. Find the minimum value of $\sum_{k=1}^{100} \mathrm{~L}(\mathrm{k})$.

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The 25th question: Let $\left(a_{1}, a_{2}, \ldots, a_{100}\right)$ be a permutation of $\{1,2, \ldots, 100\}$, and let $L(k)$ be the largest $m$ such that there exist $k=i_{1}<i_{2}<\ldots<i_{m}$ with $a_{i_{1}}>a_{i_{2}}>\ldots>a_{i_{m}}$ or $a_{i_{1}}<a_{i_{2}}<\ldots<a_{i_{m}}$. Find the minimum value of $\sum_{k=1}^{100} \mathrm{~L}(\mathrm{k})$.
For the permutation $\left(a_{1}, a_{2}, \ldots, a_{100}\right)$, let $f(k)$ denote the length of the longest increasing subsequence starting with $a_{k}$, and let $g(k)$ denote the length of the longest decreasing subsequence starting with $a_{k}$. Then, $L(k) = \max \{f(k), g(k)\}$.
It is easy to see that these 100 ordered pairs $(f(k), g(k)) (1 \leq k \leq 100)$ are all distinct.
In fact, for $1 \leq i \leq j \leq 100$. If $a_{i} < a_{j}$, then $f(i) > f(j)$; if $a_{i} > a_{j}$, then $g(i) > g(j)$. Therefore, these 100 ordered pairs $(f(k), g(k)) (1 \leq k \leq 100)$ are all distinct.

Plotting these 100 ordered pairs $(f(k), g(k)) (1 \leq k \leq 100)$ as coordinates on a plane, we get 100 distinct points. It is easy to see that for any $m \in \mathbb{Z}^{+}$, the number of positive integers $k$ such that $L(k) = \max \{f(k), g(k)\} = m$ does not exceed $2m - 1$.
Noting that $100 = \sum_{k=1}^{10} (2k - 1)$, we have
\[
\sum_{k=1}^{100} L(k) \geq \sum_{k=1}^{10} (2k - 1)k = \frac{10 \times 11 \times 21}{3} - \frac{10 \times 11}{2} = 715
\]
When $\left(a_{1}, a_{2}, \ldots, a_{100}\right) = (91, 92, \ldots, 100; 81, 82, \ldots, 90; \ldots; 1, 2, \ldots, 10)$, the equality holds!