New Zealand Mathematical Olympiad Committee

Camp Selection Problems 2015 — InstructionsSolutions due date: 23rd September 2015

These problems will be used by the NZMOC to select students for its International Mathe-matical Olympiad Training Camp, to be held in Auckland between the 3rd and 9th of January2016. Only students who attend this camp are eligible for selection to represent New Zealand atthe 2016 International Mathematical Olympiad (IMO), to be held in Hong Kong in July 2016.We have been fortunate in obtaining some sponsorship for the camp so have been able to keepthe cost to $600. This covers all expenses for the camp including travel for those from outsideof Auckland. Of course, any donations in excess of this amount will be gratefully accepted andreceipts will be provided for tax purposes.

At the camp a squad of 10–12 students will be chosen for further training, and to take partin several international competitions, including the Australian and Asia-Pacific MathematicalOlympiads. The New Zealand team for the 2016 IMO will be chosen from this squad.

Students will be selected into two groups for the camp: juniors and seniors.

• If you are currently in year 12, or you have been a member of the NZIMO training squad,then you will be considered only as a senior.

• Otherwise, you will be considered as either a junior or a senior.• Since the problems and the participants vary from year to year it is hard to be preciseabout the selection criteria. However, as a rough guide, if you solve five or more of theproblems completely then you will be in the running for selection as a junior. Obviously,the criteria for senior selection are somewhat higher.

General instructions:

• Although some problems seem to require only a numerical answer, in order to receive fullcredit for the problem a complete justification must be provided. In fact, an answer alonewill be worth at most 20% of the credit for a problem.

• You may not use a calculator, computer or the internet (except as a reference for e.g.,definitions) to assist you in solving the problems.

• All solutions must be entirely your own work.• We do not expect many, if any, perfect submissions. So, please submit all the solutionsand partial solutions that you can find.

Students submitting solutions should be intending to remain in school in 2016 and shouldalso hold New Zealand Passports or have New Zealand Resident status. To be eligible for the2016 IMO you must have been born on or after 14 July 1996, and must not be formally enrolledin a University or similar institution prior to the IMO.

Your solutions, together with a completed Registration Form (overleaf), should be sent to

NZMathematics Olympiads, University of Otago, Department of Computer Science,PO Box 56, Dunedin 9054, New Zealand

arriving no later than 23rd September 2015. Please complete the registration formcarefully and legibly, in particular your contact details. We regret that we are unable toaccept electronic submissions. You will be notified whether or not you have been selected forthe Camp by 24th October 2015.

July 2015

www.mathsolympiad.org.nz

Registration FormNZMOC Camp Selection Problems 2015

Name:

Gender: male/female

School year level in 2016:

Home address:

Email address:

Home phone number:

School:

School address:

Principal:

HOD Mathematics:

Do you intend to take part in the camp selection problems for any otherOlympiad camp?

yes/no

If so, and if selected, which camp would you prefer to attend?Have you put your name forward for a Science camp or any other camp inJanuary?

yes/no

Are there any criminal charges, or pending criminal charges against you? yes/no

Some conditions are attached to camp selection. You must be:

• Born on or after 14 July 1996• Studying in 2016 at a recognised secondary school in NZ• Available in July 2016 to represent NZ overseas as part of the NZIMO team if selected.• A NZ citizen or hold NZ resident status.

Declaration: I satisfy these requirements, have worked on the questions without assistancefrom anyone else, and have read, understood and followed the instructions for the Januarycamp selection problems. I agree to being contacted through the email address I have supplied.

Signature: Date:

Attach this registration form to your solutions, and send them to

NZMathematics Olympiads, University of Otago, Department of Computer Science,PO Box 56, Dunedin 9054, New Zealand,

arriving no later than 23rd September 2015.

New Zealand Mathematical Olympiad Committee

Camp Selection Problems 2015Due: 23 September 2015

Notes:

• If any clarification is required, please contact Michael Albert (michael.albert@cs.otago.ac.

nz).

• There are nine problems and you should attempt to find solutions for as many as youcan. Solutions (i.e., answers with justifications) and not just answers are required for allproblems, even if they are phrased in a way that only asks for an answer.

• These are selection problems only – you will not receive a ‘score’, only an indication ofwhether or not you were selected.

• For further information, see the general instructions on the registration form that accom-panies these problems.

Questions:

1. Starting from the number 1 we write down a sequence of numbers where the next numberin the sequence is obtained from the previous one either by doubling it, or by rearrangingits digits (not allowing the first digit of the rearranged number to be 0). For instance wemight begin:

1, 2, 4, 8, 16, 61, 122, 212, 424, . . .

Is it possible to construct such a sequence that ends with the number 1,000,000,000? Isit possible to construct one that ends with the number 9,876,543,210?

2. A mathematics competition had 9 easy, and 6 di�cult problems. Each of the participantsin the competition solved 14 of the 15 problems. For each pair, consisting of an easyand a di�cult problem, the number of participants who solved both those problems wasrecorded. The sum of these recorded numbers was 459. How many participants werethere?

3. Let ABC be an acute angled triangle. The arc between A and B of the circumcircle ofABC is reflected through the line AB, and the arc between A and C of the circumcircleof ABC is reflected over the line AC. Obviously these two reflected arcs intersect at thepoint A. Prove that they also intersect at another point inside the triangle ABC.

4. For which positive integers m does the equation:

(ab)2015 = (a2 + b2)m

have positive integer solutions?

5. Let n be a positive integer greater than or equal to 6, and suppose that a1, a2, . . . ,anare real numbers such that the sums ai + aj for 1 i < j n, taken in some order,form consecutive terms of an arithmetic progression A, A + d, . . .A + (k � 1)d, wherek = n(n� 1)/2. What are the possible values of d?

6. In many computer languages, the division operation ignores remainders. Let’s denote thisoperation by //, so for instance 13//3 = 4. If, for some b, a//b = c, then we say that c isa near factor of a. Thus, the near factors of 13 are 1, 2, 3, 4, and 6. Let a be a positiveinteger. Prove that every positive integer less than or equal to

pa is a near factor of a.

7. Let ABC be an acute-angled scalene triangle. Let P be a point on the extension ofAB past B, and Q a point on the extension of AC past C such that BPQC is a cyclicquadrilateral. Let N be the foot of the perpendicular from A to BC. If NP = NQ thenprove that N is also the centre of the circumcircle of APQ.

8. Determine all positive integers n which have a divisor d with the property that dn+ 1 isa divisor of d2 + n2.

9. Consolidated Megacorp is planning to send a salesperson to Elbonia who needs to visitevery town there. It is possible to travel between any two towns of Elbonia directly eitherby barge or by mule cart (the same type of travel is available in either direction, and theseare the only types of travel available). Show that it is possible to choose a starting townso that the salesperson can complete a round trip visiting each town exactly once andreturning to her starting point, while changing the type of transportation used at mostone time (this is desirable, since it’s hard to arrange for the merchandise to be transferredfrom barge to cart or vice versa).