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With the start of this academic year's USAMTS, we are going to get a some questions that have their origin in these problems, some more closely worded to the original than others. Perhaps it is time to also consider how closely a question must match a contest question in order to fall under our contest questions policy. Note the current wording of the section on identifying contest problems:

What is a contest question?

For the purpose of this discussion, a contest question refers to a question that is

  1. originally published by a third party, for the purpose of inviting submission of solutions: this could be an actual competition where a prize of some sort is given, or this could be a qualifying examination.
  2. publicly available: the questions themselves should be publicly available.
  3. time-limited: the "contest" should be active for a fixed, finite duration of time, with a definite start and end date. Before the end date of the contest, the contest is said to be "on-going"; after the end date the contest is said to have "finished" or "expired."

Note a couple caveats:

  • If the question is not original (for example, if the math.SE question was asked before the start of the contest), then we do not consider it as a contest question. This is to prevent ex post facto stifling of discussion.
  • If the "contest" has no definite duration, then we do not consider questions on it as contest questions for this discussion. This is to prevent indefinite lock-down of information.

But it doesn't give any indication of how closely matched a question here to a contest question must be in order to fall under the policy. The purpose of this "question" is to gather some opinions in order to possibly expand the policy is something more concrete. Or perhaps this is not something that can be strictly codified, but we know it when we see it.


To attempt to begin the discussion, consider the following question from the USAMTS 2007-08 Round 1 problems:

Gene has $2n$ pieces of paper numbered $1$ though $2n$. He removes $n$ pieces of paper that are numbered consecutively. The sum of the numbers on the remaining pieces of paper is $1615$. Find all possible values of $n$.

Here is a small selection of possible restatements of this question that could appear on the site. Which do you feel should fall under our policy (ignoring that the deadline for this contest passed a long time ago)?

  1. Jean has $2k$ pieces of paper numbered $1$ though $2k$. She removes $k$ pieces of paper that are numbered consecutively. The sum of the numbers on the remaining pieces of paper is $1615$. Find all possible values of $k$.

  2. An evil wizard approached the king. "Bring me some of your top advisors and their spouses, and I will number them $1$ through $2n$. You then select $n$ consecutive people (according to this numbering) to be summarily executed. If the numbers of the remaining people do not add up to $1615$, you and the rest of these people will also be killed."

    How many advisors should the king summon to avoid being killed?

  3. Suppose that $n$ consecutive numbers are removed from the numbers $1$ through $2n$ and the sum of the remaining numbers is $1615$. What are the possible values of $n$?

  4. For which values of $n$ is there is an $1 \leq k \leq n+1$ such that $\sum_{i=1}^{2n}i - \sum_{i=k}^{k+n-1}i = 1615?$

  5. Find all pairs $(k,n)$ with $1 \leq k \leq n+1$ such that $$3n^2+3n-2kn = 3230.$$

  6. Solve the Diophantine equation $3a^2+3a-2ab = 3230$.

  7. Suppose that the sum of the numbers remaining after removing $n$ consecutive numbers from the numbers $1$ to $2n$ is equal to some fixed number $a$. How can one determine the possible values of $n$?

  8. Is there a formula for the sum of the numbers remaining after removing $n$ consecutive numbers from $1$ to $2n$?

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    $\begingroup$ My view is that whenever a Question is recognizably derived from a contest problem, it should be identified as such. If the OP has failed to do this, then a Comment should be left to notice the issue. I'd treat the first four variants as covered by the contest questions policy and ask for locking. $\endgroup$ – hardmath Sep 14 '16 at 13:21
  • $\begingroup$ I think 4-8 are acceptable and the rest are not (points for the evil wizard, though). $\endgroup$ – heather Sep 15 '16 at 11:30
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    $\begingroup$ A fair question imo would be one where OP only asks about methods one can use to solve this and that type of problem. Of course the problem with this in practice is that if OP uses the acctual contest problem as an example someone is going to provide a full solution no matter what OP writes defeating the purpose. Anyway for me only something like 7 and maybe 8 is acceptable assuming it is also clearly stated that OP do not want full solutions just general pointers (and preferably also mentions that it comes from an ongoing contest). $\endgroup$ – Winther Sep 15 '16 at 13:15
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    $\begingroup$ The first four are clearly covered. The next two are probably covered: they do show some useful work, but they also ask for the hard part of the problem. I’m inclined to agree with @Winther on the last two, and I’d be more comfortable with them if they read ‘after removing $k$ consecutive numbers’. $\endgroup$ – Brian M. Scott Sep 17 '16 at 22:09
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    $\begingroup$ @heather 7 is clearly not acceptable. $\endgroup$ – Jack M Sep 22 '16 at 22:22
  • $\begingroup$ @heather: All of them are totally not acceptable, regardless of contest policy, because they show absolutely no effort and no context. Also, all except (5) and (6) are obvious restatements of the contest problem. (Any of the contest organizers will know it when they see it!) $\endgroup$ – user21820 Dec 13 '16 at 7:49

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