2011 Camp Selection Problems – due 10th August 2011

Tuesday, June 28, 2011 11:11
Posted in category Easy, Medium, News, Problems

Our first step in choosing the team to represent New Zealand at the 2012 IMO in Argentina is to choose 24 students to attend a week long training camp in Auckland in January. These students will be chosen using the 2011 Camp Selection Problems (pdf, 111k). Students submitting solutions to the Camp Selection Problems must intend to be in school in 2012; must have been born on or after 20 July 1992; and must be NZ citizens or hold NZ Resident status.

Those selected for the camp will do 1-2 assignments prior to the camp, and we hope to have them sit Round 1 of the British Mathematical Olympiad in December. At the camp in January we will choose a squad of 10-12 students for further training, and to take part in several international competitions, including the Australian and Asia-Pacific Mathematical Olympiads. The final team of six will be chosen based on the results of these competitions, and perhaps some additional selection tests, if needed.

We look forward to receiving your solutions!

- Chris Tuffley
Leader, 2011 NZIMO team

PS: The 2011 team will be departing soon for the 2011 IMO, in Amsterdam. Watch this space for news!

Added, March 2013: Here are the solutions to the camp selection problems.

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64 Responses to “2011 Camp Selection Problems – due 10th August 2011”

  1. Ian Seong says:

    June 29th, 2011 at 8:50 pm

    In S6, is A a set of random points i.e. doesn’t have to be in shape of a rectangle?

  2. Chris says:

    June 30th, 2011 at 10:03 am

    Dear Ian,

    we would typically say that the set of points is “arbitrary” rather than “random”, but the intent is the same. You have to show that any subset of G, with at least the given number of points, has the property, no matter how it happens to be arranged.

    Best wishes,

  3. Jaehwan says:

    July 1st, 2011 at 6:32 am

    Just to clarify, for the very first question, we use the same integer in a different row? so like
    125 or 134? just a random example.can

  4. Chris says:

    July 1st, 2011 at 10:05 am

    Dear Jaehwan,

    yes, the same integer can appear in two different rows, it just can’t appear twice in the same row. So you could have 1,2,5 in one row, and 1,3,4 in another row, as you suggest.

    Best wishes,


  5. Luke says:

    July 1st, 2011 at 5:17 pm

    So when you say row in the first question, you mathematically mean a horizontal arrangement of 3 numbers?

  6. Chris says:

    July 1st, 2011 at 6:07 pm

    Dear Luke,

    just think of a noughts and crosses board, but instead of filling it in with noughts and crosses, fill it in with positive integers instead. Then the board has three rows, and each row contains three numbers. Does that clear things up?

    Best wishes,


  7. Ian Seong says:

    July 2nd, 2011 at 12:02 pm

    We all know that 4, 8, 16384 are examples of powers of 2. Is 1 also a power of 2?

  8. Luke says:

    July 2nd, 2011 at 1:03 pm

    yea, is it?? Wouldnt it be 2^0 ideally which would be 1??

  9. Ian Seong says:

    July 2nd, 2011 at 1:13 pm

    2^0 is 1, that’s why I want to know

  10. Chris says:

    July 4th, 2011 at 12:08 pm

    Dear Ian, Luke,

    yes, 1 is a power of two (and of three, and of four, and …), because as you observe 1=2^0 (and 3^0, and 4^0, and …). I’m not sure which problem this question relates to, however?

    Best wishes,


  11. Ian Seong says:

    July 4th, 2011 at 4:15 pm

    Oh lol this is related to writing solution for J6

    (I had to write “since … power of 2″, and I didn’t know whether that included 1 or not)

  12. Thomas says:

    July 5th, 2011 at 1:45 am

    I have a couple of questions:

    1) Can we type our solutions using LaTeX? Whenever I handwrite my solutions, I always end up making tons of errors and end up with crossouts all over my solutions. (I’m not sure if using LaTeX counts as “computer assistance” or not)

    2) For S6, can parallelograms be degenerate? I realize that the “parallelogram” made of the “four” vertices (1,1), (1,1), (5,5), (5,5) would probably not be a parallelogram, but could the points (1,1), (3,3), (8,8), (6,6), for example, be considered the vertices of a parallelogram? As a degenerate quadrilateral, it has equal-length opposite sides, but its diagonals intersect in a lot more than one point, so I’m not sure.

  13. Chris says:

    July 5th, 2011 at 10:35 am

    Dear Thomas,
    in answer to your questions:
    (1) You’re welcome to type up your solutions using LaTeX or other document preparation software. The restriction on computer use applies to solving the problems, not presenting your solutions. More generally, we don’t mind if handwritten solutions are written in pen or pencil – the main thing is that they be clear and easy to read.
    (2) In problem S6 the paralleolgrams must be nondegenerate (i.e. have positive area/have non-collinear vertices).
    Best wishes,

  14. Frank says:

    July 6th, 2011 at 9:56 pm

    Is it acceptable to use a computer software to present my diagrams (which I already know the appearance of) in a neater, cleaner printed format for me to submit my answers with and use to work out problems? I find it much nicer looking and helpful to me if I can print my diagrams off, because the lines and circles are more precise, and I can scribble in them with pencil (I can’t do this if I have to handdraw them because I can’t make circles drawn with pen). I’ve always used Paint to draw out my diagrams although I have used Wingeom before.

  15. Chris says:

    July 7th, 2011 at 6:49 pm

    Dear Frank,

    we would encourage you to draw your diagrams by hand: this is what you have to do in a competition setting, and so being able to draw good diagrams by hand is an important skill to have. However, for the selection problems we will accept diagrams drawn using a computer, provided the capabilities of the software used are sufficiently limited.

    The use of any drawing package specifically designed for geometry problems is not acceptable, as these can help you solve the problem (for example, by dynamically showing you how the configuration changes as you drag points about). Any software used should only be capable of static diagrams, and should be used simply as “electronic straightedge and compass” : that is, it should just be used to do the things you can by hand, which is draw circles and straight lines.

    Since there are many more drawing packages out there than we can reasonably be familiar with, I’d rather not answer questions about whether a specific program is or is not acceptable. Knowing the capabilities of your software, please make your own judgment of whether your use of it would be considered assistance.

    (Incidentally, if you want to be able to scribble on your diagrams in pencil, and then erase the pencil without erasing the diagram, you could consider drawing a diagram by hand, making some photocopies, and then working on the copies.)

    Best wishes,

  16. George Han says:

    July 8th, 2011 at 4:40 pm

    Can we scan our solutions and sent them to you? Thanks

  17. William says:

    July 8th, 2011 at 4:43 pm

    Sorry to ask, but how detailed do you want our solution’s explanations to be? Will we be considered incapable of being concise if we attempt to cover every base (explain everything)?

  18. Chris says:

    July 9th, 2011 at 9:33 pm

    Dear George,
    this year solutions may be submitted electronically, provided they are sent as a PDF file. Please convert any typed or scanned work to PDF before emailing it, to ensure that we are able to read it – remember that we may not use the same software as you.
    The answer to this question may be dfferent next year, depending on who is marking the selection problems, so please check again next year if you wish to email your solutions.
    Best wishes,

  19. Chris says:

    July 9th, 2011 at 9:46 pm

    Dear William,
    we won’t be judging you on how concise your solution is, but rather on how correct it is, and on the whole I think we’d rather see you explain too much rather than too little. So more detail is generally better than less.
    Having said that, there’s no need to include proofs of any standard facts that you use – for example, if you want to use the fact that if a+b is even, then a and b are either both even or both odd, you can treat it as something known and use it without proof.
    Best wishes,

  20. Ted says:

    July 12th, 2011 at 8:55 pm

    Hi. In S6, does the parallelograms have to have at least one pair of sides perpendicular to the x or y-axis? For example, does (2,1), (1,3), (3,5), (4,3) count as a parallelogram in this problem? Thanks.

  21. Chris says:

    July 21st, 2011 at 11:23 am

    Dear Ted,
    sorry for the slow reply, I couldn’t reply while at the IMO jury for reasons of exam security. The parallelograms are not required to have a pair of sides perpendicular to either axis, so your example does count as a parallelogram for this problem.
    Best wishes,

  22. Bronson says:

    July 23rd, 2011 at 6:05 pm

    Hi, I have a question about S6, to clarify, when you say “Show that there are at least 2011^2 parallelograms whose vertices lie in A”. Does this mean that all four vertices of each parallelogram must lie in A? Or is it just that one of the vertices of each parallelogram must lie in A?. And for the next part, when all the diagonals meet at a single point, must this point be common to all the 2011^2 parallelograms or just a point within the subset G?. Thank you.

  23. Chris says:

    July 23rd, 2011 at 10:16 pm

    Dear Bronson,
    all four vertices of each of the 2011^2 parallelograms must belong to the given set A. If you look at any given parallelogram then its diagonals will intersect, and this point of intersection must be the same for all of the 2011^2 parallelograms. However, there’s no condition on this point of intersection – it doesn’t have to belong to either A or G.
    Best wishes,

  24. Bronson says:

    July 25th, 2011 at 6:10 pm

    Thank you for your assistance, Chris

  25. Timothy says:

    July 25th, 2011 at 10:09 pm

    If I would like to scan my solutions and send them via email, what is the address that I should send it to?

  26. Chris says:

    July 27th, 2011 at 12:14 pm

    Dear Timothy,
    the address you should use is the one for Michael Albert given on the registration form: michael.albert@cs.otago.ac.nz .
    As noted in the comments above please send your work as a pdf file.
    Best wishes,
    Chris Tuffley

  27. Frank says:

    July 30th, 2011 at 12:25 pm

    Hi Chris,
    Sorry to bother you again, but could I possibly email my solutions to you guys at the email address stated above AND post them to you as well? I am a bit paranoid at one method failing me. Of course this will run the risk of you guys receiving two sets of solutions which might be marked independently and considered as two different people. Maybe this could be avoided by having a notice at the front of my set of solutions stating that the solutions have been both emailed and posted. Would it be a bit too much of a hassle? Which method would you prefer?

  28. Chris says:

    July 30th, 2011 at 5:35 pm

    Dear Frank,

    I don’t think it would be a problem if you submit your solutions by both email and ordinary mail, if you make it clear on both submissions that you are submitting it by both methods.

    If you are concerned to make sure that your work arrives, you could consider keeping a photocopy or scan of it, and posting your work by a method that lets you track it. That way you can be sure it’s arrived.

    Best wishes,

  29. Frank says:

    July 31st, 2011 at 7:02 pm

    Truly sorry about the spamming of questions! Hopefully this will be the last one…
    I’m planning to sit the Chemistry Olympiad Training Group questions later this year: http://www.chem.canterbury.ac.nz/Olympiad/, so am I required to mention that on my entry form for this? For example, the question: “Are you planning to take part in the camp selection problems for any other Olympiad camp?”. I have never sat that particular test before but from what I can see, this camp and the training for Chemistry Olympiad don’t clash; their training sessions are in the first term and in April. So do I need to worry about the question: “Which camp would you prefer to attend?”, because it seems not applicable to my situation?

  30. Chris says:

    August 1st, 2011 at 12:44 pm

    Dear Frank,
    if you plan to sit the Chemistry Olympiad selection test as well do please mention that on your registration form, and indicate a preference. Although the training camps may not clash, there is an agreement among the NZ science olympiads that students can only take part in the training programme for one Olympiad at a time – so if you are selected for both the mathematics and chemistry training programmes, you will still only be able to attend one of them.
    There are two reasons for this policy. Firstly, there is a lot of work involved in training for a science olympiad (particularly in the later stages), and we do not believe that a student training for two olympiads will be able to do justice to both. Secondly, each programme can only accommodate a limited number of students, and by insisting that students take part in at most one, we give more students the opportunity to take part.
    Please note that you can take part in the programmes for different olympiads in different years – just not two in the same year.
    Best wishes,

  31. Frank says:

    August 2nd, 2011 at 8:37 pm

    Thanks Chris. Since the training programme selection for Chemistry will be determined after this camp selection (it is in November), IF I am selected to participate in the Maths training programme I will still not be sure if I will be successful for the chemistry olympiad training. I am not sure whether I will be able to ‘change’ from one olympiad to the other since I do not know about possible acceptance into the Chemistry Olympiad as it is held after this one. Will an acceptance into this prohibit me from sitting the test for Chemistry? Will I be allowed to decide later, after results for both have been announced, and in the unlikely but possible event that I am accepted into both, which one I will then be inclined to take? And finally, will a mention of a preference for Chemistry on my registration automatically or partially hinder my application for this? Could you please give me some suggestions? I personally want to give both a try, at least in the application stage. Thank you!

  32. Joy says:

    August 3rd, 2011 at 1:09 am

    Can we scan the solutions and send them by email instead of by post? If yes, which email do we send it to?
    Thank you!

  33. George Han says:

    August 3rd, 2011 at 8:35 pm

    Chris, if the quality of the scans is not very good, should we send you a printed copy of the solutions as well?

  34. Chris says:

    August 3rd, 2011 at 11:23 pm

    Frank – I will consult and get back to you by email.

    Joy – yes, you can scan your solutions and email them in, see comments above. The address to use is the one for Michael Albert, as he is marking the selection problems this year.

    George – if the scan is of poor quality, and you can’t improve it by changing the settings and re-scanning, then either post your solutions in as well, or post them in instead. If you submit your work twice by different methods, please indicate on both submissions that you are doing so.

    Best wishes,


  35. Young says:

    August 6th, 2011 at 1:49 pm

    Hi Chris, for question J3, do you mean that it is DEFINITELY possible to work out the strongest and second strongest players after 18 matches or that there may be, in some cases, a chance for it to be possible to find the strongest and second strangest players after 18 matches? O_o

  36. William says:

    August 6th, 2011 at 3:07 pm


    It’s definitely possible to find them in 18 matches (and order them).
    That’s all I’m saying :)

  37. Young says:

    August 7th, 2011 at 7:54 pm

    haha… thanks William! I got it yesterday : P also, Chris, just a question. I’m planning to email the questions in except its going to be a really really big PDF file so can I send it through in about 5-6 emails with each email containing a small section of all my answers? (I have no idea how to compact my scanned answers) or do you not mind me sharing my answers with you on Google Docs? 

    Thank you : )

  38. Jerry says:

    August 7th, 2011 at 10:44 pm


    Can I just clarify for J1, when it says rows it does not include columns? As in like there will only be three sets of three numbers, rather than six sets of three numbers with each individual number contributing to two sets?


  39. Chris says:

    August 8th, 2011 at 2:42 pm

    Young – if you wish to email your answers, please send only a single email with a single PDF file. If you are unable to do this then please post your answers instead.

    Jerry – that’s correct, there is no condition on the columns of the square, just the rows.

    Best wishes,

  40. Thomas says:

    August 12th, 2011 at 7:53 am

    Will there be some sort of confirmation that our solutions have been received? I sent my solutions in ten days ago via email; I just would like to make sure they were received.

  41. luke says:

    September 30th, 2011 at 7:26 pm

    Sorry, I would just like to know whether the people that have been selected have already been notified? Im just curious as to when the results will be announced

  42. Chris says:

    October 2nd, 2011 at 11:48 am

    Sorry, we’re a bit later announcing the results than we’d hoped. Invitations to those selected will be going out shortly.
    Best wishes,

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    New Zealand Maths Olympiad Committee online » 2011 Camp Selection Problems – due 10th August 2011

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