<s>[INST] Given reals $a_1, a_2, \cdots, a_{n - 1}, a_n$ such that $\sum_{i = 1}^na_1^2 = 1$. Calculate the maximum value of $\sum_{cyc}|a_1 - a_2|$. 
Given reals $a_1, a_2, \cdots, a_{n - 1}, a_n$ such that $a_1^2 + a_2^2 + \cdots + a_{n - 1}^2 + a_n^2 = 1$ $(n \in \mathbb N, n \ge 3)$. Calculate the maximum value of $$\large |a_1 - a_2| + |a_2 - a_3| + \cdots + |a_{n - 1} - a_n| + |a_n - a_1|$$

There must exist $1 < k < n$ such that $a_{k - 1} \le a_k \le a_{k + 1}$
$ \implies |a_{k + 1} - a_k| + |a_k - a_{k + 1}| = |a_{k + 1} - a_{k + 1}|$.
Repeat the above process for about $n - 1$ times and we have that $$|a_1 - a_2| + |a_2 - a_3| + \cdots + |a_{n - 1} - a_n| + |a_n - a_1| \le 2 \cdot \min(|a_i - a_j|, 1 \le i < j \le n)$$
Now we just have to find the maximum value of $\min(|a_i - a_j|, 1 \le i < j \le n)$ for $$a_1^2 + a_2^2 + \cdots + a_{n - 1}^2 + a_n^2 = 1$$, which I don't know how.
 [/INST] We have that $|x - y| = 2 \cdot \max(x, y) - (x + y)$
$$\implies \sum_{cyc}|a_1 - a_2| = 2 \cdot \left[\sum_{cyc}\max(a_1, a_2) - \sum_{i = 1}^na_1\right]$$
which could be rewritten as $$\sum_{cyc}|a_1 - a_2| = 2 \cdot \sum_{i = 1}^nx_ia_i$$ where $x_i \in \{-1, 0, 1\}, i = \overline{1, n}$ and $\displaystyle\sum_{i = 1}^nx_i = 0$.
In the case of $n$ being odd-numbered, there must exist $m$ $(1 \le m \le n)$ such that $x_m = 0$, otherwise $\displaystyle\sum_{i = 1}^nx_i$ would be odd.
Let $x_n = 0$, we obtain that $$\sum_{cyc}|a_1 - a_2| \le 2 \cdot \sum_{i = 1}^{n - 1}x_ia_i \le 2 \cdot \sum_{i = 1}^{n - 1}|a_i| \le 2\sqrt{(n - 1) \cdot \sum_{i = 1}^{n - 1}a_i^2} = 2\sqrt{n - 1}$$
The equality sign occurs when $a_i = \pm \sqrt{\dfrac{1}{n - 1}}, i = \overline{1, n - 1}$ and $a_n = 0$ such that $\displaystyle\sum_{i = 1}^{n}a_1 = 0$.
The same progress could be done for even-numbered values of $n$.
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