Medal allocation

[Prateek Karandikar]

The medal allocation rules from http://www.ioi2010.org/rules.shtml
aren't completely clear.

\begin{quote}

After the second Competition Day and before the IOI Awards Ceremony
the medal distribution is determined by an automatic procedure, based
on the number of points the contestants achieved. The medal awards are
uniquely determined by the following rules:

1. The score necessary to achieve a gold medal is the largest score
such that at least one twelfth of all contestants receive a gold
medal.
2. The score necessary to achieve a silver medal is the largest
score such that at least one fourth of all contestants receive a
silver or a gold medal.
3. The score necessary to achieve a bronze medal is the smallest
score such that at most one half of all contestants receive a medal.

\end{quote}

Suppose all contestants get a score of, say, 226. The "score necessary
to achieve a gold[/silver] medal" seems to be 226. The bronze cutoff
is more confusing. The set of potential bronze cutoffs which ensure
that at most half the contestants get a medal seems to be the open
interval (226, infinity). The bronze cutoff is apparently the minimum
of this set, which does not exist. What medals will the automatic
procedure allocate in this case?

The 3 conditions
1. At least 1/12 of the contestants get a gold medal.
2. At most 1/2 of the contestants get a medal.
3. Two contestants with the same score get the same medal, or both get
no medal.

are inconsistent. (The inconsistency is not likely to arise in
practice, but the rule talks of uniquely defining medal allocation,
without making exceptions for unlikely cases)

Overall, the rule seems quite vaguely stated anyway and makes several
unstated assumptions (eg, a contestant gets a gold iff their score is
>= the score obtained by bullet point 1). If the intention is to be
completely precise, the rules should be stated in a way fit for an IOI
problem statement of the type "you are given N contestants' scores.
[insert medal rules here]. Output for each gold, silver, bronze, or no
medal".

If the intention is not to handle all possible cases, this should be
stated explicitly, possibly with something like "corner cases will be
handled on a case by case basis".

[gvc]

The medal allocation rules have been used exactly as stated for
several IOIs.

If more than 1/2 of the contestants tied for best score, the formula
would indeed award (gold) medals to more than 1/2 of the contestants,
contrary to IOI regulations.

This situation will not occur.

I'm not aware of any other cases in which the formula might be
inconsistent.