Tag Archives: UKMT

RMM 2015 – UK Team Blog Part Two

Posted on by
Reply

Saturday 28th Feburary

There is much to squeeze into the programme, and so the second paper starts an hour earlier. James and I spend the day based at Tudor Vianu, the specialist maths and computing school that has hosted this competition since its first incarnation in 2008. Even coming from schools which regularly send students to such international competitions, we find it remarkable to see how much explicit emphasis they place on academic excellence here. Where British observers might expect lists of prefects and photos of glorious football teams, here instead we see students posing with medals and Romanian flags from contests around the world.

Our first task is to coordinate yesterday’s problems. Qs 1 and 3 are agreed extremely rapidly, with the problem captains very complimentary of the UK boys’ number theory solutions. Q2 has all the ingredients to suggest a long slog, but coordinators Lucian Turea and Radu Gologan have clearly thought very carefully about the UK scripts. Everything they say is sensible and easy to make consistent, so we are finished and happy comfortably within our 20 minute slot.

James and I also have to supervise the Romanian scripts for Qs 2 and 3 as these are British submissions. My schoolboy Latin helps a little bit, and we eventually agree on how to pair up comments in the solutions with points on the markscheme just in time to meet our students as they finish. Joe is full of excitement, having completed all three in the closing moments, while others are disappointed, having been slightly thrown by the geometry, but overall spirits are high.

The team are squirrelled off by the guides, and I have the afternoon to engage with the Q5 scripts. We have five solutions, and all are fine, but might not appear fine to a casual observer. Liam has opted for the Jackson Pollock approach to truth, where statements of various levels of interest and veracity are independently sprayed freely across three pages, though after a while I am convinced that every line does follow from something somewhere else on the page.

While working, I realise that having a ground floor room in an Eastern European hostel has its drawbacks. That said, opening the window gives the chance for an experiment to determine exactly which genre of music is found least appealing by lingering smokers. Enescu’s sonate dans le caractere populaire roumain proves successful, despite its local heritage.

I have a better idea what’s going on in all our students’ arguments in time to venture down to the slightly baffling Bucharest metro towards the farewell dinner, which retains its name despite not falling on the final evening. The students are deliberately separated from the leaders, but no attempt is made to enforce this and everyone mingles freely. This year the Chinese team comes from the Shanghai area, and their leader teaches at their high school. He has recently spent a term in Reading and, together with Warren and Harvey, we have a highly enthusiastic conversation about differences in education systems between our countries.

The UK students’ table seems to have been chosen for especially ponderous service, but the 30 seconds they are given between their desserts arriving and the bus arriving proves sufficient. I feel judged when I arrive back at my room and find the Hungarian deputy leader still working on his problematic geometry, so make sure to have at least a nominal further glance at our Q5s before setting an early alarm.

Sunday 1st March

The only people in Victory Square at 6.30am are stray dogs, and stray leaders heading to the school to prepare for their coordinations. I’m apprehensive about being asked to go through Harry’s solution to Q5 line-by-line, but though the effort to understand everything felt purposeful, it wasn’t necessary, as we get what we request almost immediately. Exactly the same thing happens on the other second day questions, so James and I are kicking our heels by 9.30, and the possibility of a return to bed feels very inviting while we wait for the other countries’ scores to clear.

Joe write: Meanwhile, we are at the mysterious ‘Hostel X’, in order to visit an ‘escape room‘. This was not actually as dubious as it first sounds. As Dominic explained, we were to be locked in a room for an hour during which we would have to solve a number of puzzles in order to escape. [DY: think of The Crystal Maze but with less leopard-print.] This turns out to be extremely enjoyable, as we gradually discover the collective significance of some masks, a chessboard and a couple of UV torches, but also quite difficult. Sam, Harvey, Andrei and I manage to escape with barely five minutes to spare but the others do not quite finish, although they assure us that their room was by far the more difficult of the two…

We meet the students, who are discussing Morse decoding and similar things with great enthusiasm. En route home, James thinks that we have been too hasty to accept a flaw in Joe’s solution to Q6, since it suddenly dawns that it could be fixed with the addition of a single \ge sign. Our original coordinators have gone home, but chief coordinator Mihai Baluna graciously takes a second look, and agrees with our re-assessment, so James’ defibrillator can go back in the box.

The bronze and silver medal boundaries have settled naturally, and after brief discussion the jury decides to round up the number of golds to ten, which is surely the right decision. This leaves the UK with three honourable mentions, two silvers, and a gold. Though some of our students might be disappointed to lie just below a boundary, they all recognise that this contest features challenging problems and experienced contestants, many from countries with far more strenuous training programmes than ours. By any measure, this is a fantastic team performance, and James and I are very proud of them.

The closing ceremony is held in the atrium of Vianu school, and after an encouraging speech from the headteacher, the medals are awarded fairly swiftly. Joe reports that the hardest aspect of winning a gold at RMM is the necessity to smile on stage continuously for three minutes. Russia is announced as the winner of the team competition, with a very impressive set of performances, closely followed by the USA.

With the entire evening clear, the UK and USA teams head to Piata Romana to celebrate each other’s successes. The Romanian guides and the UK leadership have slightly different views about what constitutes an appropriate venue for this, but in the end everyone is entirely happy to gather in the common area at James’ hotel. This has been an excellent competition, and it is wonderful to see students, guides and leaders from all teams finding so much in common and much to learn from one another.

Monday 2nd March

My roommate departs for a train to Budapest at 4am, and the accommodation staff are enthusiastically dismantling the bunkbeds in the adjacent room at 6am, so it is fair to say I might have slept better. Two cars take us out to the airport, precisely one of which thinks we are having a race through the rush hour traffic. Suffice it to say, I would probably like to cycle round the Arcul de Triumf even less than its Parisian counterpart.

The students’ recently acquired metalwork doesn’t quite take us over the baggage weight limit. The Wizzair boarding procedure leaves a little to be desired, but the party looks keen for little except sleep, so it makes no difference to do this sparsely. As ever, the arrivals barriers at Luton only just manage to hold back the legions of adoring fans. Goodbyes are exchanged before we head our separate ways, though we will meet again for more worthwhile mathematics in just over three weeks at our next training camp in Cambridge.

RMM 2015 – UK Team Blog Part One

Posted on by
Reply

The UK was invited to send a team to the Romanian Master of Mathematics competition, held in Bucharest for the seventh time in 2015. This is a short account of what happened. There are moments when I wasn’t present, for which Joe reports on behalf of the students.

A pdf version with more details about the organisation of the contest, statements of the problems and a brief summary of results can be found here. Background and reports on similar competitions can be found here, including links to more comprehensive registers of hosted elsewhere.

Wednesday 25th February

We have spent the night at a hotel close to Luton Airport, so we can proceed to our flight on foot. Walking in front of a bright red sunrise to a bright orange terminal to depart on a bright pink plane leaves me with a sense of colour overload not experienced since I last watched South Pacific. The three hour flight to Bucharest is unremarkable. Sam has fallen asleep with pencil poised halfway through a long expression where every other term is 2012^{2012}, and Harvey makes rapid progress on a dodecahedronal Rubik’s Cube.

Soon afterwards we arrive in Romania and get lifts to the Moxa accommodation complex of the University of Economics where the students will stay. There are clearly mild cultural differences concerning what levels of privacy middle-aged adults might expect to enjoy, but the organisers have done a good job, and all is resolved satisfactorily. Of the students, Harry and Sam will share with two Brazilian boys who are due to arrive in the middle of the night, and the remaining four have a dorm to themselves, complete with precarious looking upper bunks.

Joe writes: Slightly surprisingly, we have been given seven guides, an entire Year 11 class at the Vianu school. Four of them take us on a walking tour round the central area of Bucharest, including the imposing Victory Square, and Herastrau Park. A sign informs us we should not toboggan down the brief slope between the path and the lake. We heed this advice.

Later in the evening, James and I diverge for the first leaders’ meeting. Old friendships are renewed, and the proceedings are informal and brief, allowing as much time as possible to get to grips with the questions. A proposed pair of papers is circulated by chief problem selector Ilya Bogdanov, and we get to work in James’ room. Our immediate impression is that we like them a lot, and this is reaffirmed over the coming hours as we explore them further.

Thursday 26th February

The leaders get to work finalising the papers. My confidence in the quality of the questions has grown even stronger overnight, and so I am not surprised when these are approved fairly rapidly. I propose swapping questions 3 and 5 based entirely on my own prejudice regarding their relative difficulty, and it turns out that others feel similarly, and this is approved.

Next, we must finalise a definitive wording of the questions before they are translated into languages for 14 other countries. Various people have strong views on commas, how many times one should use the word `let’ in a given sentence, and whether `open’ or `interior’ are more likely to be found ambiguous by students. Perhaps unexpectedly, a question submitted by the UK, courtesy of Lex Betts, causes the most problems for wording. In the end, it seems easiest to avoid ambiguity by completely rephrasing it in terms of the blackboard setup that will now appear as Q3 on the paper, accompanying Q2, the work of our own Jeremy King.

Joe writes: Meanwhile we enjoy a more comprehensive tour of Bucharest, past the old city and the Palace of the Parliament, then on to an excellent lecture by Calin Popescu. He tells us about topological dimension, and we learn that triangles are two-dimensional, though unsurprisingly the real challenge is deciding precisely what `triangle’ and `two-dimensional’ actually mean.

The opening ceremony is a well-organised affair in the grand university hall with several generous speeches from the mayor and other local dignitaries, and representatives of Tudor Vianu school. On the way home, the students examine their goodie bags, featuring various stationery and an RMM polo shirt. The leaders have not been missed out, though I wonder whether their guesses at sizes may have been informed by my predecessors? Certainly I will have to eat a lot of the omnipresent potato salad to run any danger of fitting into this item before the end of the competition…

After our winter camp in Hungary, the students are now connoisseurs of Eastern European cuisine, and remain unfazed by even the most remarkable display of gherkins. While James and I catch up on work, they relax before tomorrow’s festivities by starting another round of the card game which I am apparently not allowed to name. Suffice it to say, it has a similar quality to The Archers, offering a nonsensical background murmur which proves surprisingly supportive to research productivity.

Friday 27th February

Harry reports over breakfast that he spent some of the night helping dismantle a hyperactive burglar alarm, but it seems everyone is feeling well-prepared for the first day of the contest. James and I have carefully assembled a selection of fruit for the UK team’s refreshment, but, after Snow White themed questions regarding our intentions, the apples prove substantially more popular with the Hungarian students.

After approving answers to a handful of questions, mostly about the nature of the `first turn’ in Q2, we are free, so I return for a walk around the serpentine Herastrau Lake. The boundary of the lake seems to have Hausdorff dimension slightly greater than 1, but in any case, it is pleasant to stop halfway round its seemingly infinite perimeter to work on some problems about multitype branching processes. I also stop at the orthodox cathedral, from which my own college chapel could learn plenty about how a solemn space can be gold without being gauche.

Our students seem unsure whether to be upbeat or not, but we have a complete set of solutions claimed for Q1, and some cautious reports of progress on Q2. To avoid wasting time worrying about the recent past, some of the guides scoop up the Russian, American and UK teams for a walking tour of Bucharest old town and the stylish Cismigiu park. As in 2008, I observe that Bucharest enjoys a surfeit of excellently-equipped playgrounds almost everywhere, but a total absence of children using them. On this occasion, the younger members of our team are reluctant to rectify this.

I get started on the Q2s after dinner, and in a pleasing reversal of what often happens at some competitions, our two students claiming partial solutions have actually done rather better than they suggested. Sam in particular has been very clear about what he can and cannot do, and might even end up scoring seven. James and I convene at his hotel to discuss the challenging Q3 which seems to be equally clear-cut, so it is a hard-working evening, but a lot less drawn-out than it might have been. It is good to see that our recent active efforts to encourage the students to improve their write-up style are paying dividends.

IMO 2013 – Part One: Travel and Training

Posted on by
1

Preamble

Six years ago in Rhodes, Tom Lovering and I started what has now become a strong tradition of preparing an unofficial report about maths competitions from the student perspective. It seems appropriate to attempt to continue this in my new role as the Deputy Leader of this year’s UK team at the IMO. And since I have excellent wifi and a (just about) active maths blog, there seems no reason not to do this in real time, at least to a first approximation. I’m sure fans all around the world will be glued to their screens.

I should briefly explain what the IMO is. The acronym stands for International Mathematical Olympiad, and it is a competition held every year in July, welcoming school students from over 100 countries. Tempting though it is to picture a drawn-out global version of the ‘mathletics’ scene at the end of Mean Girls, the reality is somewhat different. Each country sends six students, who sit two 4.5 hour exams, each with three questions, in roughly increasing order of difficulty. It does however remain the case that you get jackets if you make the finals, admittedly with polyester rather than leather sleeves. Medals are awarded to roughly half of the participants.

Each team has a leader, who arrives early to help set the paper, and also assesses their team’s scripts, presenting their marks for approval by a board of co-ordinators supplied by the host country. Each team also has a deputy leader, who stays with the team initially, then joins the leader for this marking process.

As well as the competitive side, the olympiad is a great opportunity to meet other young mathematicians from all around the world. Certainly I am still in touch with many of the people I met when I was lucky enough to compete in Vietnam and Madrid (2007, 2008 respectively). As the competition moves country every year, it’s also a great chance to see some exciting places. This year it is in Santa Marta on Colombia’s Caribbean coast, after Buenos Aires in 2012.

Anyway, on with the report.

Sunday 14th July

I spend the morning packing up my room as I am moving to a new flat pretty much directly after this trip. Everything seems a lot clearer after sorting out the IMO team uniform which has arrived just in time leaving my floor essentially invisible under a sea of boxes. The mini-crisis wherein they were all delivered without my knowledge to the Worcester College kitchens seems but a distant memory…

We are flying at a painfully early hour tomorrow morning, so it makes sense to spend the night at an airport hotel. Courtesy of the satnav, I learn the hard way that there are three Holiday Inns at Heathrow. Geoff, Bev and I are the first to arrive, and wait for the students, two of whom are arriving directly from Copenhagen, bearing prizes and stories from the analogous physics competition just finished there. Parents are reassured that the occasional email and postcard will be sent and we retire in preparation for tomorrow’s Odyssey.

Monday 15th July

Up at 4.30am for the first leg over to Madrid. With time for little other than a quick espresso, straight onto the transatlantic flight to Bogota. The ten hours afford plenty of time to catch up on reading some papers. Had a think about how these results about (uniform) random forests might affect our thoughts about frozen percolation, and took advantage of the increasing tedium to do a long rate function calculation I’d been putting off for ages. I think the answer is \frac{1}{2}(1-\frac{1}{\lambda}) – the question is somewhat more interesting…

Also relish the chance to spend several hours diving into Love in the Time of Cholera, having figured that this was almost certainly a once-in-a-lifetime opportunity to explore a Colombian novelist while in Colombia. So far, so good. In particular, much more interesting than One Hundred Years of Solitude, or perhaps my tastes have changed in the past few years?

We learn courtesy of Iberia that tuna, peach and olives do not make a good sandwich combination, and wonder whether they will be able to resist the temptation to follow every announcement with a synthesised rendition of the Concierto de Aranjuez. A slight delay changing at Bogota airport allows sufficient time for extra sushi and further progress through the example sheet solutions I’ve offered to \LaTeX before the short hop north to Santa Marta. Gabriel’s cynicism about the fate of our luggage turns out to be unfounded, but the two panama hats packed in my suitcase have not enjoyed the trip at all. The Santorini Hotel seems ill-prepared for a group arrival at 10.30pm, but eventually we obtain keys and pay. Shortly afterwards, we are able to unpay one of the bills that they have charged us twice within the space of five minutes. With everyone very grateful for the violent air conditioning, we head for much overdue sleep.

Tuesday 16th July

Up at dawn from the jetlag, but a useful moment to sort out the details for the first practice exam. This pre-IMO camp is a joint venture with the Australian team, and both sets of students are sitting an IMO style exam each morning. The villa we are occupying is somewhat sort on table space, but the three UK students perched on the kitchen bar with their scripts claim that it is fine. If IMO 2008 is anything to go by, where the desks for the competition were so steeply sloped that pens became more valuable as paperweights than as writing equipment, this might be useful practice.

While the students are getting on with the festivities, Bev and I explore various local food options, I study a couple of papers and explore the beach, though the humidity is rather cloying in the middle of the day. The UK team make confident noises about the exam, so I hope that marking the Q2 geometry won’t be too traumatic. Some complicated diagram dependencies render this hope in vain, but we finish up in time for a quick debrief before dinner. Meanwhile, the team have learned the hard way that Colombian plumbing does not hugely appreciate toilet paper…

Wednesday 17th July

I would normally struggle rather badly to find the motivation to go for a 7am run, but with a mile or so of relatively quiet beach on offer, it suddenly becomes a much more attractive proposition. As I return to the Santorini resort, the first waves of peddlers are arriving. One or two make a token attempt to sell me sunglasses, and a nice lady asks me how I got a particularly purple bruise, though I figure my Spanish is not sufficient to explain the idea of cricket right now.

Geoff bids us farewell and heads off to join the other team leaders at a top-secret location where they will begin the process of setting the paper. In theory it’s top-secret; in practice, it must be Barranquilla, the next city down the coast. The students power through another exam all morning, and pleasingly resist the temptation to make anything too complicated, so marking everything is relatively straightforward. Our stroll to dinner is accompanied by a small posse of feral dogs. I am reminded of the health guidance for this part of the world: “rabies is relatively low-risk, except for children, who are more likely to allow themselves to be licked in the face.”

Related articles

Modular arithmetic – Beyond the Definitions

Posted on by
Reply

Modular arithmetic is a relatively simple idea to define. The natural motivation is to consider a clock. The display of a standard analogue clock makes no distinction between 4am, 4pm, and 4pm next Thursday. This is a direct visualisation of the integers modulo 12. Instead of counting in the usual way, where each successive integer is different to all those considered previously, here, every time we get to a multiple of 12, we reset our count back to zero. As a result, this procedure is often referred to as `clock arithmetic’.

A common problem for good students, for example those starting the UKMT’s Senior Mentoring Scheme, is that the idea of modular arithmetic seems very simple, but it’s hard to work out how it might be useful in application to problems. My claim is that the language of modular arithmetic is often the best way to discuss divisibility properties in problems about whole numbers. In particular, the behaviour of powers (ie m^n and so forth) is nice in this context, and the notation of modular arithmetic is the only sensible way to approach it. Anyway, let’s begin with a quick review of the definitions.

Definitions

We are interested in divisibility by some fixed integer n\geq 2, and the remainders given after we divide by n. Given an integer M, we can write this as:

M=kn+a,\quad\text{ where }a=0,1,\ldots,n-1,

and this decomposition is unique. We then say that M is congruent to a modulo n. Note that working modulo n, means that we are interested in remainders after division by n (rather than a or k or M or anything else). This has the feeling of a function or algorithm applied to M. We get told what M is, then work out the remainder after division by n, and say that this is `M mod n‘.

This is fine, but it very much worth bearing in mind a slightly different interpretation. Working modulo n is a way of saying that we aren’t interested in the exact value of an integer, only where it lies on the n-clock. In particular, this means we are viewing lots of integers as the same. The `sameness’ is actually more important in lots of arguments than the position on the n-clock.

More formally, we say that a\equiv b or a is congruent to b modulo n, if they have the same remainder after division by n. Another way of writing this is that

a\equiv b\quad \iff \quad n|a-b.

Sets of integers which are equivalent are called congruence classes. For example \{\ldots,-4,-1,2,5,8,\ldots\} is a congruence class modulo 3. Note that under the first definition, we consider all elements here to be congruent to 2, but in a particular question it may be more useful to consider elements congruent to -1, or some combination.

These definitions are equivalent, but it can be more useful to apply this second definition for proving things, rather than writing out b=kn+a or whatever all the time.

Addition and Multiplication

Everything has been set up in terms of addition, so it is easy to see that addition works well on congruence classes. That is:

\text{If }a\equiv b,c\equiv d,\quad\text{then }a+c\equiv b+d.

We could argue via a clock argument, but the second definition works very well here:

\text{We have }n|a-b,n|c-d,\quad\text{and so }n|(a+c)-(b+d),\text{ exactly as required.}

We want to show that a similar result happens for multiplication. But this holds as well:

\text{If }a\equiv b,c\equiv d,\quad\text{then }n|c(b-a)\text{ and }n|b(c-d).

\Rightarrow n|ac-bd,\text{ that is }ac\equiv bd.

Let’s just recap what this means. If we want to know what 157\times 32 is modulo 9, it suffices to say that 157\equiv 4 and 32\equiv 5, and so 157\times 32\equiv 4\times 5\equiv 2. In a more abstract setting, hopefully the following makes sense:

\text{ODD}\times\text{ODD}=\text{ODD};\quad \text{EVEN}\times\text{EVEN}=\text{EVEN};\quad \text{ODD}\times\text{EVEN}=\text{EVEN}.

This is exactly the statement of the result in the case n=2, where the congruence classes are the odd integers and the even integers.

Powers

What happens if we try to extend to powers? Is it the case that

\text{if }a\equiv b,c\equiv d,\quad\text{then }a^c\equiv b^d? Continue reading →