New Zealand Mathematical Olympiad Committee

Camp Selection Problems 2012 — InstructionsSolutions due date: 15th August 2012

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 6th and 12th of January2013. Only students who attend this camp are eligible for selection to represent New Zealandat the 2013 International Mathematical Olympiad (IMO), to be held in Colombia in July 2013.The cost of the camp is yet to be determined precisely, but will not be more than $500.

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 2013 IMO will be chosen from this squad.

There are two sets of problems: junior division and senior division.

• If you are currently in year 12, or you have been a member of the NZIMO training squad,then you may only attempt the senior problems.

• If you are currently in year 11 or below, and you have never been a member of the NZIMOtraining squad, then you may attempt both sets of problems (and your results from bothsets will be taken into account in the selection process).

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 only be worth 1 point out of 7.

• You may not use a calculator, computer or the internet (except as a reference for e.g. def-initions) 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 2013 and shouldalso hold New Zealand Passports or have New Zealand Resident status. To be eligible for the2013 IMO you must have been born on or after 12 July 1993, 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

Dr Christopher Tu!ey, Institute of Fundamental Sciences, Massey University Man-awatu, Palmerston North 4442

arriving no later than 15th August 2012. We regret that we are unable to accept electronicsubmissions. You will be notified whether or not you have been selected for the Camp by 7thOctober 2012.

If you have any questions, please contact Chris Tu!ey (c.tu!ey@massey.ac.nz, (06) 356 9099x3573) or Michael Albert (michael.albert@cs.otago.ac.nz).

July 2012www.mathsolympiad.org.nz

Registration Form

NZMOC Camp Selection Problems 2012

Name:

Gender: male/female

School year level in 2013:

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 12 July 1993• Studying in 2013 at a recognised secondary school in NZ• Available in July 2013 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

Dr Christopher Tu!ey, Institute of Fundamental Sciences, Massey University Man-awatu, Palmerston North 4442,

arriving no later than 15th August 2012.

New Zealand Mathematical Olympiad Committee

Camp Selection Problems 2012Due: 15 August 2012

Junior division

J1. From a square of side length 1, four identical triangles are removed, one at each corner,leaving a regular octagon. What is the area of the octagon?

J2. Show the the sum of any three consecutive positive integers is a divisor of the sum oftheir cubes.

J3. Find all triples of positive integers (x, y, z) with

xy

z+

yz

x+

zx

y= 3

J4. A pair of numbers are twin primes if they di!er by two, and both are prime. Prove that,except for the pair {3, 5}, the sum of any pair of twin primes is a multiple of 12.

J5. Let ABCD be a quadrilateral in which every angle is smaller than 180!. If the bisectorsof angles !DAB and !DCB are parallel, prove that !ADC = !ABC.

J6. The vertices of a regular 2012-gon are labelled with the numbers 1 through 2012 in someorder. Call a vertex a peak if its label is larger than the label of its two neighbours, anda valley if its label is smaller than the label of its two neighbours. Show that the totalnumber of peaks is equal to the total number of valleys.

Senior division

S1. Find all real numbers x such that

x3 = {(x+ 1)3}

where {y} denotes the fractional part of y, i.e. the di!erence between y and the largestinteger less than or equal to y.

S2. Let ABCD be a trapezoid, with AB ! CD (the vertices are listed in cyclic order). Thediagonals of this trapezoid are perpendicular to one another and intersect at O. The baseangles !DAB and !CBA are both acute. A point M on the line sgement OA is suchthat !BMD = 90!, and a point N on the line segment OB is such that !ANC = 90!.Prove that triangles OMN and OBA are similar.

S3. Two courier companies o!er services in the country of Old Aland. For any two towns,at least one of the companies o!ers a direct link in both directions between them. Addi-tionally, each company is willing to chain together deliveries (so if they o!er a direct linkfrom A to B, and B to C, and C to D for instance, they will deliver from A to D.) Showthat at least one of the two companies must be able to deliver packages between any twotowns in Old Aland.

S4. Let p(x) be a polynomial with integer coe"cients, and let a, b and c be three distinctintegers. Show that it is not possible to have p(a) = b, p(b) = c, and p(c) = a.

S5. Chris and Michael play a game on a 5 " 5 board, initially containing some black andwhite counters as shown below:

Chris begins by removing any black counter, and sliding a white counter from an adjacentsquare onto the empty square. From that point on, the players take turns. Michael slidesa black counter onto an adjacent empty square, and Chris does the same with whitecounters (no more counters are removed). If a player has no legal move, then he loses.

(a) Show that, even if Chris and Michael play cooperatively, the game will come to anend.

(b) Which player has a winning strategy?

S6. Let a, b and c be positive integers such that ab+c = bcc. Prove that b is a divisor of c, andthat c is of the form db for some positive integer d.