I have a probabilistic solution for Problem A (optimised) https://codeforces.net/contest/2068/submission/308681793

E is, supposedly, well known in romania

H when the distances are the same. https://atcoder.jp/contests/abc135/tasks/abc135_e

Btw A can be solved with $$$4n - 7$$$ votes: 308693499.

This can be proven inductively. WLOG, assume $$$m = \binom{n}{2}$$$. The $$$n=2$$$ case is trivial. For $$$n \ge 3$$$, from the inductive hypothesis, say we have a set $$$V$$$ of $$$4n - 11$$$ votes getting the desired facts concerning only the candidates $$$1, 2, \ldots n-1$$$. Now, let $$$S$$$ be the set of candidates who we want to beat $$$n$$$ and $$$T$$$ be the set of candidates who we want to be beaten by $$$n$$$. Let $$$\sigma_S$$$ be any ordering of $$$S$$$ and $$$\sigma_T$$$ be any ordering of $$$T$$$. For $$$2n-6$$$ of the votes from $$$V$$$, make $$$n$$$ the most preferred candidate, and for the remaining $$$2n-5$$$ votes, make $$$n$$$ the least preferred candidate (keeping their ranking among $$${1, 2, \ldots n-1}$$$ the same). Let $$$\hat{\sigma_S}, \hat{\sigma_T}$$$ denote the reverse of $$$\sigma_S, \sigma_T$$$ respectively.

Now, we add four more votes, with the following rankings:

1. $$$\sigma_S, n, \sigma_T$$$
2. $$$\hat{\sigma_S}, n, \hat{\sigma_T}$$$
3. $$$\sigma_T, \sigma_S, n$$$
4. $$$n, \hat{\sigma_T}, \hat{\sigma_S}$$$

Intersetingly, $$$O(n/ \log{n})$$$ votes suffice and are asymptotically optimal.

Interestingly, I came up with pretty much same problem as D. Morse Code for another contest some years ago, it got rejected (reason: it appeared in some shortlist very long time ago, $$$O(N^4)$$$ intended solution IIRC)

Later I discovered $$$O(N^2)$$$ solution in a paper.

I have used ide one so it may be the reason for coinciding the solution