## 2013 Camp Selection Problems – due 20th October 2013

Tuesday, September 3, 2013 10:03Our first step in choosing the team to represent New Zealand at the 2014 IMO in Cape Town, South Africa, is to choose 24 students to attend a week long training camp in Auckland on 12-18 January 2014. These students will be chosen using the 2013 Camp Selection Problems (pdf, 182kb). Students submitting solutions to the Camp Selection Problems must intend to be in school in 2014; must have been born on or after 10 July 1994; and must be NZ citizens or hold NZ Resident status.

We hope to have most of those selected for the camp sit Round 1 of the British Mathematical Olympiad on 30 November. At the camp in January we will choose a squad of 10-12 students for further training, and to sit the Australian and Asia-Pacific Mathematical Olympiads and Round 2 of the BMO. The final team of six will be chosen based on the results of these competions and several assignments, and perhaps some additional selection tests, if needed.

We look forward to receiving your solutions!

- *Chris Tuffley*

Leader, 2013 NZIMO team

**Note:** We regret that we are unable to accept electronic submissions of solutions to the camp selection problems. All submissions should be sent in hard copy to the address given on the registration form.

Richard Zhou says:

September 3rd, 2013 at 8:53 pm

Hi, for question two of the camp selection problems would negative numbers be considered prime numbers, e.g. -3

Thanks

Chris says:

September 4th, 2013 at 3:42 pm

Dear Richard,

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means that negative numbers such as -3 are not considered to be prime.

One reason for excluding negative numbers is to give us uniqueness of prime factorisation. If we allowed negative numbers to be considered to be prime, then for example 6 would have two prime factorisations: 2×3 and (-2)x(-3).

Best wishes,

Chris

Sesein Sean says:

September 5th, 2013 at 7:01 pm

Hi, for question 5, f from whole number to the whole number, does it mean that f:A->A for the group A is that whole number?

Daniel says:

September 5th, 2013 at 8:30 pm

Just to reiterate/rephrase Sesein’s question, does question 5 mean that any whole number put into the function will return a whole number?

Wen Gao says:

September 6th, 2013 at 9:05 am

Is question 8 weird on purpose, or am I missing something here?

Chris says:

September 6th, 2013 at 9:49 am

Daniel, Sesein: If f is a function from the whole numbers to the whole numbers, then f takes whole numbers as inputs, and returns whole numbers as outputs. So, for any whole number x, you can plug x into f, and the result f(x) will also be a whole number; and if x is not a whole number, then f(x) is not defined, meaning that you can’t plug x into f.

Wen: I don’t understand what you’re asking. You’ll have to rephrase your question if you’d like a meaningful answer.

Best wishes,

Chris

Sesein Sean says:

September 6th, 2013 at 7:40 pm

Can international students take part in this event?

Chris says:

September 6th, 2013 at 8:27 pm

Dear Sesein,

Unfortunately only NZ citizens and residents are eligible for camp selection. This is because the purpose of the camp is to begin choosing and training the IMO team, and under the IMO regulations, contestants are expected to be citizens or residents of the country they represent.

Best wishes,

Chris

Isabel says:

September 8th, 2013 at 11:19 am

Is question number 9 correct when it says R must be inside the triangle?

Wen Gao says:

September 8th, 2013 at 12:12 pm

Nevermind, question 8 is fine. I was being stupid

Daniel Baard says:

September 8th, 2013 at 9:25 pm

Hi there, do you take into account how long it took for someone to submit their answers? i.e. the time taken to answer all of the questions.

Sally says:

September 9th, 2013 at 5:55 pm

What does question 3 want us to find?

Chris says:

September 9th, 2013 at 6:50 pm

Isabel: If you think about the condition on angles CAB and CBA, you’ll see that it’s possible to choose R so that it lies inside ABC.

Daniel: We have no way of knowing how long someone took to answer the problems, so no – we only take into account the quality of their solutions. If your question is “Is there an advantage to submitting my solutions earlier?”, then no: all submissions received by the deadline are treated equally.

Sally: This question isn’t exactly asking you to find something, but rather to give a proof that a certain thing is possible. The reciprocal of the number m is the number 1/m, so the question is asking you to prove that for n>2 we can always express 1 as a sum of n different numbers of the form 1/k, where each k must be a positive integer. It’s obvious that we can write 1 as a sum of reciprocals if we’re allowed to use the same number more than once (for example 1/2 + 1/2 = 1, 1/3 + 1/3 + 1/3 =1, etc); the challenge is to prove that for n>2 we can still do it even if we aren’t allowed to repeat a number.

Best wishes,

Chris

Christopher says:

October 1st, 2013 at 9:26 am

For question 5 is 0 considered to be a whole number?

Jasmine says:

October 1st, 2013 at 9:37 pm

Are we allowed to use a picture of an octahedron for question 4? Also, I used a calculator for some of the questions, without reading the instructions, but I only used it for simple things so that I could do things more quickly. Is that OK, of should I redo the questions?

Anggie says:

October 2nd, 2013 at 12:42 pm

I checked the definition for whole number on Wikipedia, and it says:

Whole numbers may variously refer to:

natural numbers beginning 1, 2, 3, …; the positive integers

natural numbers beginning 0, 1, 2, 3, …; the non-negative integers

all integers …, -3, -2, -1, 0, 1, 2, 3, …

So…in question 5, does ‘whole number’ include 0 and negative numbers?

Chris says:

October 2nd, 2013 at 2:28 pm

Christopher, Anggie: as stated in the question, “whole numbers” is used here to mean the non-negative integers, which consist of 0,1,2,3,… So 0 is included, but the negative integers -1,-2,-3,… are not.

Jasmine: You’re welcome to use a picture of an octohedren in your answer for question 4. It does however always pay to read the intstructions before jumping into something…It would be best to redo the questions where you used a calculator.

Best wishes,

Chris

Minho says:

October 2nd, 2013 at 9:06 pm

Hi, I just wanted to clarify for question 5, is it asking to find other functions that share the same properties listed, excluding all (different) functions that have the result of 1 for any whole number and if not, prove why not?

Also, it says that the forms must be sent by post arriving no later than 20th October, do I just have to post it early in case it doesn’t get posted asap?

Sidney says:

October 21st, 2013 at 3:09 pm

When do you expect to complete your evaluations and send out the notices of results? Will you publish answers at that time?

Thank you

Angela Shi says:

October 23rd, 2013 at 4:45 pm

Hi, I sent my submission through courier and I was wondering if it has been received since I did not receive a confirmation email?

Thanks

futC7goC says:

August 30th, 2015 at 7:54 am

775633 759050 Spot on with this write-up, I truly believe this site needs significantly a lot more consideration. I

credit reports from all three bureaus says:

January 2nd, 2016 at 1:09 pm

3755 134525I