An interesting contest math problem: find the maximum value of $f(a_1,a_2,...,a_n)$

You are asking to maximize

$$\begin{equation}\begin{aligned} f(a_1,a_2,...,a_n) & = \vert a_1-a_2\vert+\vert a_2-a_3\vert+...+\vert a_{n-1}-a_n\vert \\ & = \sum_{i=1}^{n-1}\vert a_{i} - a_{i+1} \vert \end{aligned}\end{equation}\tag{1}\label{eq1A}$$

If $a_i \ge a_{i+1}$, then $\vert a_i - a_{i+1} \vert = a_i - a_{i+1}$, else $\vert a_{i} - a_{i+1} \vert = a_{i+1} - a_{i}$. In either case, when you remove the absolute value signs to get the corresponding equivalent value, you have one sequence term being added and another one subtracted. Thus, over the $n - 1$ absolute value terms, you would have $n - 1$ sequence terms being added and $n - 1$ sequence terms being subtracted.

Note that $a_1$ and $a_n$ are used just once each. Also, $a_i$, for $i \le 2 \le n - 1$, are being used twice. The maximum possible value of \eqref{eq1A} would occur if all of the $n - 1$ sequence terms being added were the largest ones and with the $n - 1$ sequence terms being subtracted being the smallest ones, and with the special case of the $2$ terms of $a_1$ and $a_n$, as they are used just once, being the $2$ largest terms among those being subtracted.

This can be done. In particular, as stated above, have the $2$ "middle" (with ones at either side if $n$ is even) values be at the start and end of the sequence. Then at the odd indices from the start, i.e., $3,5,7,\ldots$ and the even indices back from the end, i.e., $n-2,n-4,n-6,\ldots$, you alternate back & forth placing the remaining smallest values from the smallest up. Also, at the even indices from the start, i.e., $2,4,6,\ldots$, and the odd indices back from the end, i.e., $n-1,n-3,n-5,\ldots$, you alternate back & forth placing the largest values from the largest down.

The top "half" of the values will always be beside a value from the bottom "half" of the values on either side of them in the expression in \eqref{eq1A} and, thus, the top half values will be the ones added and the bottom half values will be the ones subtracted. I'm using "half" in quotes because the $2$ ones near the middle are handled specially and there's a small issue with odd versus even values of $n$ to deal with.

As you say you used to be a contest math lover as a high school student and have majored in math now, I trust you can finish the rest yourself.