The Balkan Mathematical Olympiad is a competition for secondary school students organised annually by eleven countries in Eastern Europe on a rotating basis. The 2018 edition was held near Belgrade, Serbia over the period 7-12 May 2018. The UK was grateful to be invited as a guest nation.

Our participation is arranged by the UK Maths Trust, as part of a broader programme to introduce the country’s most enthusiastic young mathematicians to regular problem-solving, challenging mathematics, and several annual opportunities to participate in competitions. For the Balkan MO, we have a self-imposed rule that students may attend at most once, so that as many as possible might enjoy the experience of an international competition.

The non-geometry problems of the contest are discussed at length in this blog post, and the geometry problem which appeared as Q1 is discussed at considerable length, along with some background on harmonic ranges, in this blog post. A full report encapsulating all these aspects, is available here.

This post covers the non-mathematical aspects of the contest, which was enjoyed by all the UK students.

**Problem selection**

The programme of this competition is a scaled down version of the IMO. The leaders gather in suburban Belgrade on Monday night to select four problems from a shortlist compiled by the organisers. To recreate the students’ experience, it makes sense to start by trying these without reference to solutions. Some of the questions are UK submissions, so I can briefly astonish my colleague Vesna with almost instant fluency, before admitting that I wrote or edited the corresponding solutions.

Making the choice occupies Tuesday morning. As always, it feels slightly like a shot in the dark, as one night is not really sufficient to get a feeling for twenty problems, especially the hardest ones. In the end, there was clearly a unique good hard problem, but unfortunately it had to be rejected because it was too similar to a recent problem from a well-known source. Some of us have been investing considerable energy in finding natural Euclidean arguments to the geometry problem chosen as Q3, but once Greek leader Silouanos outlines the role of harmonic ranges, it is hurriedly moved to Q1. I think the resulting set of four questions are attractive, but with a rather compressed difficulty range, and certainly not in the right order for the UK students, whose geometric toolkits probably don’t yet include the ideas needed to access the `easy’ solutions.

In any case, it’s interesting to discuss with the leaders from some of the eleven Balkan full member countries. Our opinions differ concerning which styles of problem give an advantage to extensively-trained problems. I personally feel that Q2 and Q3 are accessible even to students (or adults!) without much mathematical background, whereas here is a prevailing view that no problem with combinatorial flavour is ever ‘easy’. By contrast, many of the ideas required for a short solution to either Q1 or Q4 might be considered obscure even by serious olympiad enthusiasts, though feature on the school curriculum, at least for the most able children, in many of these countries.

We have to finalise the wording of the problems, and there are many many proposed improvements to Q2 and Q3. The final problem, unsurprisingly, requires considerably less attention. That’s our job done for the British delegation, while the other leaders get to work producing versions in their own languages, including Bosnian and Serbian, the (non-)differences between which can happily fill one dinner’s worth of interesting conversation.

**The contest**

On Wednesday morning, we are transferred to the contestant site, in the rolling hills just outside the south-east city limits of Belgrade. An extremely brief opening ceremony takes place in a room slightly smaller than the number of people attending the competition. The UK team look happy enough perched on a table. Two local violinists play Mozart with a gypsy flourish, before Teodor von Burg, a former Serbian olympiad star and graduate of Exeter College, Oxford, speaks briefly about the usual cliches of such speeches, and the additive paradox of wishing *everyone* good luck before a competition, then ends rapidly to avoid indulging such cliches himself.

After the contestants fan out to various exam rooms spread through the hotel, the contest begins and they are allowed to ask queries about the problems for 30 minutes. Many many students ask ‘what does *exactly the same route* mean?’ and ‘what if Alice and Bob play forever?’, but some variety is provided when Aron shares his detailed dilemma about the exact usage of carbon paper. (FAO future UK students: this is not to become a habit, please…)

After Monday’s 2am start, I am overdue a nap. There has been some room-swapping, and mine is reserved for ‘Professor Mr Jill Parker’. Whomever the bed truly belongs to, I leave it in time to meet the team outside the exam with Jill and Vesna. As we’d predicted, many are enthusiastic about Q2 and Q3, but have been frustrated by the geometry. Tom crowd-sources an investigation to recover a result about the incentre claimed by Alex, who perhaps now regrets, in his rush to move to other questions, not offering more of such details himself. No-one claims anything beyond observations in the number theory, so we suggest they keep thinking about it through the afternoon.

**A brief excursion**

Agnijo and Nathan had done their research on Belgrade, and had asked about the possibility of visiting the Nikola Tesla museum. The team have a guide, Sandra, a maths undergraduate, and I’m extremely impressed that she and some of her colleagues are able to organise a visit downtown and guided tour of this museum at essentially no notice for them, along with Italy, Bosnia and Azerbaijan. Vesna and I diverge to make a start on marking in a cafe, rejoining in time for the museum, where Giles apparently learns what ‘Azerbaijan’ is, and we all learn about Tesla’s extraordinary life story, and get to see the original Tesla coil (briefly) in action. Agnijo and Tom have been primed with fluorescent tubes, which do indeed glow as lightning surges between the century-old coil and its crowning sphere. Other exhibits, including highlights from Tesla’s wardrobe (pre-dating \emph{geek chic}, it would seem), and an imitation ticket from Belgrade to New York, are perhaps less fascinating.

But the roar of Volts is still in our ears as we stroll across the city centre, where Alex confidently identifies several churches as the orthodox cathedral they’d visited earlier, and eyes are drawn to the faded but strident protest banners outside the parliament. We choose a restaurant in bohemian Skadarska street, where prices are low, and availability of protein and itinerant accordion players is high. The team are trying to be polite about their hotel’s food, but I sense this variation is welcome. Giles pokes gingerly at a deep-fried pork slab, which erupts with multiple cheeses. The ‘*Serbian sword*‘ could be retitled ‘as many meat items on a stick as possible (plus 1/8 of a pepper)’.

We return to Avala feeling sleepily satisfied. Tom and Agnijo discuss the GCSE question ‘prove using algebra that the product of two odd numbers is odd’, and whether you can or should prove it without algebra. The taxis clearly sense our post-prandial vulnerability, and operate a creative attitude to receipts, and to powers of ten. But this round of ambiguous paperwork and mathematical corrections is just the prelude for Vesna and myself, who have a cosy night in with the scripts.

**Coordination**

At a competition, the leaders of each team study their own students’ work, and agree an appropriate mark with a team of local coordinators. The UK has an easier workload: we do not have to provide translations, since our students write in English, though some of them might like to note that in a question about parity, mixing up the words `odd’ and `even’ as if flipping a coin does make it harder to convince the reader you know what you’re talking about.

We start with 9am geometry, where the coordinators are proposing giving Aron 8 or 9 out of 10 as part of a crusade against citing configurational properties as ‘well-known’. Aron has, in fact, outlined a proof of his (fairly) well-known fact, and if the proposal is to award 6 or 7 without this, then the marking team’s entire day is guaranteed to be a continuous series of wars. I think the penny drops shortly after our meeting, and Aron gets upgraded to 10/10 at 9.30. Unfortunately, what remains of the crusade will deny Alex any credit at all for his unjustified claim about the incentre, despite its role in an appealing synthetic solution.

The middle two questions have a wide range of arguments. The British work on Q3 is actually pretty good, and even in the two scripts with small corners missing have organised their cases very clearly, and the coordinators (who initially want to give all full marks) can see that the students already had the ingredients to fix their minor errors. Q2 is more challenging. Once we have worked out where the good bit begins, Nathan’s solution is clearly superb, and once we’ve worked out which of his mysterious side-comments to ignore, Giles has all but the final details of a really imaginative solution, and Agnijo is flawless. Aron seems keen to make an *even* number of really confusing mistakes on this paper, so on this question has mixed up ‘horizontal’ and ‘vertical’ as if flipping a coin, though the coordinators are more sympathetic than I would have been. Tom claims that his solution is ‘very poorly written’, which is very far from the case, but after rolling back and forth through his logic a few times, we agree that a couple of cases of *q* are inadvertently missing.

The students return from their short excursion in time to hear their scores before dinner, and though Alex is a bit disappointed about the non-acceptance of his ‘lemma’, everyone is broadly pleased with themselves, as they should be. I get my first experience of the infamous hotel salad, which the students had previously described as ‘vinegar topped with lettuce’, which is roughly accurate, though the rest is nice enough. Agnijo is worried the main course includes beef, but is satisfied with the supposedly *vegan* alternative, namely a grilled fish.

The Balkan countries take the table of scores a bit more seriously than we do, and so this year’s celebratory table is sipping Bulgarian cognac washed down with Romanian tears, though this wholesome rivalry shouldn’t distract from the hugely impressive seven perfect scores from those countries’ contestants (plus four from the others). The competition at the adjacent table seems to be the relative merits of Serbian, Macedonian and Montenegrin wine and *rakija*. Meanwhile, the UK students have made plenty of new friends to induct into their favourite card games, and some Albanians, Bosnians and Turks seem a) very keen to practise their excellent English, and b) appropriately baffled by the rules, and lack of rules.

**Round and about**

The bulk of Friday is set aside for an excursion. Our destination is Valjevo, a town two hours’ drive west of Belgrade, which represents some sort of historical home for the Serbian maths enrichment community. We gather in their gloriously rococo hall to listen to an in-depth presentation concerning many aspects of daily life at Valjevo Grammar School. The nearby research institute in leafy Petnica offers a more science-focused perspective. The students get to tour some labs, though they don’t get to practise for their upcoming A-level or Highers physics by trying any experiments. Nathan, however, finalises his solution to Q4 from the contest, which seems a good use of time, and which you can read earlier in this report. Aron asks me to solve what seems a challenging geometry question in my head. I cannot. A stamp-sized freehand diagram on a napkin doesn’t help either.

Vesna was a regular visitor to Petnica as a teenage olympiad contestant, and she has briefed me on the charms of a nearby cave, apparently a regular choice for planned and unplanned excursions during her selection camps in the 90s. The UK group plans to sneak away from the third phase of the tour to find this cave, but we are foiled because the third phase of the tour is indeed a visit to the cave. This involves a short walk, during which Agnijo is harassed by the world’s least threatening dog. The temperature is pushing 30C, but Aron is worried about sunburn, so is reluctant to remove his polar fleece. He gently roasts, while Alex tells us some horror stories from his experience as a Wimbledon ballboy during the 2016 heatwave. The cave provides cool relief, and is indeed giant, with plenty of sub-caves underneath the looming stalactites.

It turns out we are in the less impressive half of the cave. The students want to climb to the more impressive upper cave. It may be more impressive, but it is also considerably darker, and I admire Giles’ and Nathan’s tenacity to find out exactly how far a distant rocky staircase extends into the gloom temporarily illuminated by a phone torch. That concludes the adventure, and we return to Belgrade coated in varying quantities of cave detritus. The return journey affords great views of the distant mountains towards the Bosnian and Montenegrin borders, though Tom is keen to use the time to make a start on coordinating the multi-author student report. Unable to avoid eavesdropping on the discussion, sounds like it will be a substantial document when completed…

**Finishing up**

Back in Avala, the closing ceremony takes place during dinner, and is informal. Jury chair Zoran Kadelburg awards the certificates; chief organiser Miljan presents the medals; and Miljan’s wife notices and steps into the essential role of helping the medallists flip their newly-acquired prizes in front of any flags they might be carrying for the waiting photographers. This one-at-a-time low-key arrangement was actually very nice for everyone, and our four medallists enjoyed their moments.

It is a balmy evening, so we drift outside again. Aron is random-walking, hunting for the WiFi sweetspot so he can download the punchline to our colleague Sam’s claimed complex solution to Q1 before Nathan finishes rounding up new players for the next round of card games; while Giles and Alex disappear off towards the most distant unlit car park with a troupe of guides and Bosnians and a volleyball. At the leaders’ table, Vesna and the other Balkan residents give a collective hollow laugh on hearing that I have elected to travel to the Montenegrin Alps by bus. But that ten hour experience starts tomorrow, outside the remit of this report, which will end here, with some pictures of mountains.