International Maths Olympiad 2011

The International maths olympiad is an annual competition for high school mathematcains who compete in national teams. This year the event is in the Netherlands. The contestants have now completed their 6 question test and can relax while they wait for the results.

Terrance Tao who is a Fields medalist and expert solver of IMO type problems has been running a mini-polymath feature on his blog each year and the third one starts today, in case you want to join in the fun. He will choose the hardest problem and invite you to find solutions, so being less ambitious I will take a look at the easiest, which is problem number one.

Given any set  A = {a1,a2,a3,a4} of four distinct positive integers, we denote the sum a1+a2+a3+a4 by sA. Let nA denote the number of pairs ( i, ) with 1 ≤ i < j ≤ 4  for which aaj divides sA. Find all sets A of four distinct positive integers which achieve the largest possible value of nA.

You might want to try to solve this before you read any further.

Here is a curious geometric way to construct the sets with maximum nA

Take a platonic solid with V vertices, E edges and F faces, take  A = {1, F-1, V-1, E-1}. Then sA = V+E+F -2 = 2E by Euler’s polyhedron formula.

a1+a2 = F  but for a regular solid the number of faces multiplies by the number of sides of each face must equal twice the number of edges, so  a1+a2 divides sA.

a1+a3 = V  but for a regular solid the number of vertices times the number of edges that meet at each vertex is also twice the number of edges so a1+a3 divides sA.

a1+a4 = a2+a3 = E which obvious divides sA = 2E

So for these sets nA ≤ 4

from the cube and octahedron we get A = {1,5,7,11} and for the dodecahedron and icosahedron we get  A = {1,11,19,29}. The tetradron gives  A = {1,3,3,5} but this is disallowed because the numbers must be distinct. You can check that nA = 4 in each case.

Other sets with  nA = 4 can be constructed by multiplying these two solutions through by a common factor. Your job is to show that this provides all solutions, and that nA = 4 is the maximum.

Update 20-July-2011: Tao choose problem number 2 for the mini-polymath. It was a geometric/combinatoric question about a “windmill” (very appropriate for Holland of course). The solution requires an observation that there is an invariant that is not immediately obvious to everyone. Meanwhile Lubos who took part in IMO 1992 has looked at problem number 5 , see his solution. (My olympiads were 1977 and 1978 by the way) If you are really interested in this type of problem you should visit the mathlinks forums where all the questions were already posted and solved long before any of us bloggers had looked at them.


15 Responses to International Maths Olympiad 2011

  1. Luboš Motl says:

    Dear Phil, I hope that my solution – added to TRF, you may also add a link – is correct. Kasuba or Kasuha or what’s the name was closest and had the right template of the key step, although he neglected some permuted possibilities which required another step – the function’s being even…

  2. carla says:

    Well I feel a lot better that both Phil and Lubos were on the IMO teams and so likey to be able to solve these sort of equations in their sleep, whereas people like me would need a week of constant concentration to get any where.

    Phil, I thought most people on the UK IMO team ended up at Trinity Cambridge whereas you ended up at Glasgow university.

    And you also had the Field’s medal winner Richard Borcherds on your team who won gold in that year.

    • Philip Gibbs says:

      I did go to Cambridge, as did everybody on the IMO team I think. We were all in different colleges but frequently met at the math society meetings. Richard was at Trinity. It was funny that he ended up using string theory because he had always said that applied math and physics was of no interest to him. He has changed his mind now. Sometimes we discuss by email. Glasgow was where I did my PhD

    • Luboš Motl says:

      Dear Phil,

      why would you called Borcherds’ stuff “applied math” or “applied physics”? In my opinion, it is indeed as pure maths as you can get. The fact that string theory is a part of it doesn’t change it, it’s still pure string theory, and it’s no wonder that string theory has overlapped with pure maths in recent decades. But that doesn’t make it “applied”.

      We also don’t say that the proof of Fermat’s Last Theorem is “applied math” because it applies the knowledge about elliptic curves. ;-) “Applied” means that it’s used for the economic benefits of people who don’t care about the maths or physics itself, isn’t it? So Borcherds’ promise that he would stay in “pure maths” was obeyed as much as I can imagine.

      Cheers, LM

    • Philip Gibbs says:

      I did not say that his work was applied math or physics, it was very much pure math. But the physics of string theory was a crucial lead to find the proof, so it was not as useless as he imagined. If string theory had not already been around the mathematicians would have had to invent it to solve the moonshine problems. I wonder how long it would have taken them.

    • Luboš Motl says:

      Did he say that “[all] physics and applied math” was of no interest to him, or “applied math and applied physics” was of no interest?

      These two things are a bit different. But for both of these interpretations, it’s still legitimate to say that even the stringy papers about the same things are “pure maths”.

      The fact that physical reasoning and concepts that are usually used in physics for some reasons are helpful doesn’t change anything. It’s true for many other things in pure maths, too, isn’t it? For example, in geometry, they measure distances in metric spaces – and distances are a typical physics thing, aren’t they?

      We call the people who invented the logic for embedding the moonshine etc. into string theory “physicists” but this is pure convention and it doesn’t prevent me from calling their beautiful results “pure maths”.

      I don’t think that Borcherds was saying that he would avoid everything that may also appear in physics because this would be preposterous and impossible. He just said that he wouldn’t be interested in physics or applied maths as the purpose of his reasoning, right? And I think he obeyed it.

      It seems conceivable that mathematicians would also discover the structure of string theory – or much of it – within a decade or a few. All roads lead to string theory. It would still be true that the mathematicians with some kind of physics intuition would be the key to make progress in some issues that were done by physicists in the real world. But in such an alternative world, even you could still call it “pure maths”.

      The point I really want to make is that string theorists are “pure” as well when it comes to the question whether applications – in ordinary Joe’s life – are a driver. They’re not.

      Cheers, LM

    • Philip Gibbs says:

      Dear Lubos,

      I agree with your description of how we now see the interplay between maths, physics and applied sciences.

      However, back in 1980 when I was at Cambridge with Borcherds there was a very different culture among pure mathematicians. They were hanging on to the ideas portrayed by the Cambridge mathematician Hardy in his book “Mathematicians Apology” . They liked to consider their work as completely disconnected from the real world and would boast that subjects like number theory would never have any applied applications whatsoever.

      Even then at Cambridge we had pure maths and theoretical physics taught as one subject under the same degree course. Students specialized one way or the other from year two but you could still take lectures on general relativity and number theory if you wanted to. Pure mathematicians were still holding on to the idea that they could mostly ignore physics, but now that has changed. The discoveries of people like Witten and Borcherds have changed that perception and now every pure mathematician knows he cant afford to ignore what physics tells us, not just about our universe but also about the structure of pure mathematical logic.

    • Luboš Motl says:

      Dear Phil, thanks for your memories. Well, there’s no full consistency in the logic, so I won’t be able to achieve a full empathy to the way how you were dividing things.

      Including Hardy to the list of the anti-physics warriors is even more bizarre than with Borcherds himself. In a recent discussion about 1+2+3+… = -1/12, Hardy was the most important guy in the history of maths who would be looking at the superficially divergent sums (such as this one) in the physical way, long before this became more standard among physicists (regularization, renormalization).

      So Hardy could have said lots of political words about separating himself from the evil physicists but in effect, his reasoning was much closer to the physicists’ reasoning than the reasoning of many other contemporaries among the mathematicians. So all these social games are often disconnected from the intellectual content.

      BTW, to get back to the thing wasting hours of ours these days, do you agree that today in the morning, we should be able to decode some more details about the recent Higgs graph you recently leaked? ;-) Do you really think that the Wednesday plenary talks will be “trivial combinations” in the sense that we will be able to reconstruct the Wednesday talks essentially today?

    • Philip Gibbs says:

      From what I have gathered, Dzero, CDF, CMS and ATLAS will each present their best all-channel combined results today. The Tevatron plenary talk will then show the important combination of Dzero and CDF with the exclusion given in the press release. The combination for CMS and ATLAS is not ready so there will be nothing extra to show at the LHC plenary, but the individual results will already be strong and interesting to compare.

      The LHC combination will be ready for LP2011. Originally ATLAS wanted to have it at EPS but CMS held it back. At the end of the year they will even combine LHC and Tevatron, but by then the LHC data will swamp the Tevatron contribution.

      That is my current best understanding, it might turn out a little different.

    • Luboš Motl says:

      Understood. You may be shown right soon.

      By the way, this blog takes 20 seconds for me to open, telling me “waiting for” or “com” most of the time. Despite your complete absence of moving widgets etc., it takes about 3 times longer for to open than e.g. TRF. Do you know that? Is it just me?

    • Philip Gibbs says:

      This one is always fast for me, but I should try it without being logged on to WordPress.

      Your’s opens quickly but then takes a long time to add extra stuff using Javascript, e.g. number of comments. Sometimes it locks completely for about 30 seconds waiting for Javascript.

    • Philip Gibbs says:

      Dzero combination paper is up

  3. Luboš Motl says:

    OK, I went through the D0 paper now. I don’t know how I could derive, from a doubling of this paper of some sort, that the Higgs is excluded outside 114-137 GeV.

    Cheers, LM

  4. Philip Gibbs says:

    Agreed, a line drawn at sqrt(2) does not give such a strong result, but the CDF plot could be better.


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