BMO1 2018

The first round of the British Mathematical Olympiad was sat yesterday. The paper can be found here, and video solutions here. Copyright for the questions is held by BMOS. They are reproduced here with permission.

I hope any students who sat the paper enjoyed at least some of the questions, and found it challenging! The following commentaries on the problems are not official solutions, and are not always full solutions at all, but contain significant steps of solutions, so would be best saved until after you have attempted the problems, if you are planning to do so. I’ve written quite a lot about Q5 because I found it hard (or at least time-consuming) and somewhat atypical, and I’ve written a lot about Q6 because there was a lot to say. I hope at least some of this is interesting to some readers of all levels of olympiad experience.

Question 1

A list of five two-digit positive integers is written in increasing order on a blackboard. Each of the five integers is a multiple of 3, and each digit {0,1,…,9} appears exactly once on the blackboard. In how many ways can this be done? (Note that a two-digit number cannot begin with zero.)

It’s a trope of BMO1 that the first question must be doable by some sort of exhaustive calculation or listing exercise. Of course, that is rarely the most efficient solution.

However, there is normally a trade-off between eliminating all listing, and reducing to a manageable task.

The key observation here is that writing the integers in increasing order is really just a way to indicate that order of the choices doesn’t matter. Even if that seems counter-intuitive. The question wants to know how many ways to choose these five numbers. The order of choice doesn’t matter since we’re going to put them in ascending order on the blackboard anyway.

You want to make your choices with as much independence as possible. So it would, for example, be a bad idea to choose the smallest number first. How many possibilities are there where the smallest number is 24? What about 42? What about 69? These are all different, and some are zero, so will make the computation very taxing.

However, you might notice that the digits {0,3,6,9} have to go together to form two numbers, and the rest have to pair up with one digit from {1,4,7} and one from {2,5,8}. You might know that an integer is divisible by 3 precisely if its digit sum is divisible by 3, but in this context you wouldn’t lose too much time by simply listing everything! These tasks are now completely separate, so you can take the number of ways to pair up {0,3,6,9} and multiply by the number of ways to pair up {1,4,7} and {2,5,8}. You need to take care over the ordering. It does (obviously) matter which is the first digit and which is the second digit in a number!

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MOG 2018

I’m teaching a lecture course at Technion this coming semester, which is going to limit opportunities for travel somewhat for the next three months or so. It was therefore nice to take advantage of the new ultra-cheap Luton-Tel Aviv connection on Wizzair to make a whistlestop weekend trip to Cambridge. It was lovely to catch up with friends, colleagues and former students, including some people who fit into all those categories.

Among my various mathematical diversions over the weekend, about thirty of us spent most of Sunday marking the 2018 edition of the UK’s Mathematical Olympiad for Girls (MOG). You can find the paper without solutions here, and the paper with official solutions and commentary here.

I have had several blog posts on probabilistic, competitive and educational topics queued up over this summer which has alternated between extreme busyness and lethargy, but this felt like a good opportunity to break the drought and both start and finish a post on the same day!

I’m going to write some thoughts about Question 4, mostly about part a), which I hope might be useful to future students who perhaps find this post while preparing for next year’s edition of the competition. Here’s the statement:

Each of 100 houses in a row are to be painted white or yellow. The residents are quite particular and request that no three neighbouring houses are all the same colour.

a) Explain why no more than 67 houses can be painted yellow.

b) In how many ways may the houses be painted if exactly 67 are painted yellow?


Of all the questions on the paper, based on the fraction of the scripts which I saw, I suspect Q4a) is the one for which the most candidates will have scored significantly fewer points than they’d been hoping for.

I think this illustrates some useful guidance for maths competitions, and also illuminates some differences between maths and other subjects which candidates might not have thought about.

Many students wrote that because three houses in a row cannot be yellow, the way to get as many yellow houses as possible is to use a YYWYYW… pattern. Since 100 is not a multiple of three, you have to work out what to do with the final house, after you’ve coloured 99 using the pattern, and sometimes you can colour it yellow, and sometimes you can’t.

Finding and proving results about maxima

Before focusing on the qualities of this argument, it’s worth returning to the question statement. It’s in two parts, and (assuming the answer to the second part isn’t “none“) then the overall goal is to find the maximum number of yellow houses, and to count the number of ways to achieve this maximum number, presumably by characterising them somehow. If the question had just asked us to find the maximum, and had not told us what it is, we would have to:

  • Decide on a value M, and show that there is a configuration with exactly M yellows.
  • Show that no value >M of yellows is possible.

Note that, apart from deciding on the value of M, which really has to come at the start of the argument, you can do these steps in either order. And if you want to find the number of configurations achieving the maximum, you have to:

  • Decide on a value M, and count the number of configurations with exactly M yellows, and check that this is positive!
  • Show that no value >M of yellows is possible.

Again, you can perform these steps in either order. If the question is posed like this (with M unknown), you have no idea as you are attempting the paper whether finding the value of M is supposed to be easy or supposed to be a really significant part of the attack on the problem, so starting your solution with a sentence like “I’m going to show that M is the maximum” and continuing by giving your proof an exoskeleton like “First, I’ll show that M+1 (or >M) is impossible. [Do this. Then leave a gap.] Now, I’ll count the number of configurations with M yellows.” This leaves no ambiguity that you are doing the problem in the right way, and for someone reading it, it might be the case that the validity of your argument is almost instantly clear once the rough outline is established.

In any case, on MOG Q4, the question has been split into two parts deliberately, and you have been told what the value of the maximum is. In fact, the order is the opposite to what I’ve proposed in the bullet points above. But this just serves to emphasise that, even when or especially when the maximum is already given, the order of attack is not important. The two bullet points might be completely unrelated mathematical arguments, but they are certainly structurally unrelated.

Structuring your approach to MOG Q4

But, to return to MOG Q4, the first part is asking you to show that 68 or more yellow houses is not possible under the rules. The second part is where you are supposed to establish what all the 67 yellow configurations are, and on this occasion it may turn out that you re-use ideas established in part a) during the enumeration of part b).

Anyway, I’m going to repeat an earlier paragraph. Many students wrote that because three houses in a row cannot be yellow, the way to get as many yellow houses as possible is to use a YYWYYW… pattern. Since 100 is not a multiple of three, you have to work out what to do with the final house, after you’ve coloured 99 using the pattern, and sometimes you can colour it yellow, and sometimes you can’t.

Now that we’ve established some general principles, we can see why some students may have got themselves into knots that weren’t directly relevant to the question.

  • The blue comment “sometimes you can colour it yellow, and sometimes you can’t” is just not relevant because at this stage we are trying to show that 68 (or more) is not possible. If at any point it becomes a discussion of whether one class of configurations leads to 66 or 67, this can’t be relevant to showing that 68 is not possible.
  • The green sentence “because three houses in a row cannot be yellow, the way to get as many yellow houses as possible is to use a YYWYYW… pattern” is, as written, true, but dangerous. Because it’s not precise. As becomes clear in part b), there are actually lots of ways to get as many yellow houses as possible, and while all of them have this pattern most of the time, it is certainly not the case that you use this pattern for the first 99, and then make a decision about the final house. (Since this only gives one configuration.)
  • The fact that 100 is not a multiple of 3 is crucial to the problem. If it had asked about 99 houses, you would have had more room just to work with a breakdown into 33 groups of three houses, and some of the argument would have been simpler. So an argument saying that 100/3 = 66.66…, which we can round up to 67 is not really explaining what’s going on. Of course we have to have a whole number of houses, but why you round up rather than down is all about the rules of colouring.
  • Finally, the notion of an ‘iterated maximum’ is a generally erroneous argument. In such an argument, you argue that the maximum way to attack a subproblem is X, and once you have X, clearly the maximal way to finish is Y, so the maximum total is X+Y. This is wrong, because of course it’s possible there might be a non-maximum way X*>X to attack the subproblem, from which you can finish in Y*<<Y, so the total ends up being lower overall.
  • In this example, the problem is not that the green statement is an erroneous iterated maximum. It’s just sounds like one. It can easily be made into a non-erroneous argument. But as soon as you start with a pattern, but then consider how to add something onto the end of it, you have essentially fallen into this trap unless you very carefully explain why taking the maximum for the first 99 houses plus the maximum for the final house must give a maximum overall.

Because the condition is a local one (meaning it tells you about rules governing two or three adjacent houses, not a global one affecting all houses in the street), it’s good to use a local argument. In the example:

(YYW)(YYW)…(YY*W)(YW*Y)…(YWY)(Y),

notice that it is not true that in every group of three consecutive houses, two are Yellow. (Look between the *asterisks*.) However, it is true that in every group of three consecutive houses, at most two are Yellow, and so in particular if we split as shown above, there are at most two yellows in all the three-brackets, and at most one at the end, giving at most 67 in total. The fact that the pattern changes from YYW to YWY does not affect this argument.

If you set it up this way, then this gives you a good headstart on part b). You might notice that the pattern can change from YYW to YWY (as in the previous example), but not the other way round. And what about WYY? Can this ever come up? You need to justify both aspects of this, and then clarify how to count the number of places where the pattern could change, having now established that all configurations have this form.

Other comments

The question setters produced a beautiful paper, and it was clear that one of their goals was to use multiple parts sensibly so as to suggest entrance routes into the problem. Many many students successfully used the factorisations discussed in the first parts of Q1, and even those who didn’t succeed seemed mostly to be trying to find differences of squares, rather than prime factors out of nowhere.

This principle applied elsewhere though, and in particular on Q3. I think some students thought Q3a) was supposed to involve three calculations, but in fact it required just giving a combinatorial justification for why the set of objects counted by b can be split into two, where one subset is counted by c and one by d. This argument, which might have been familiar to some through knowledge of Pascal’s triangle, applies equally well at other squares in the box, and also in three dimensions. If you don’t have the gap in the middle of the house, you can immediately apply this argument to Ghastly in part b). But in fact, even with the gap, you can make this work, because you are only really counting across *places you could have been just before in your path*, and the gap means that sometimes this is smaller than it would be without the gap.

There were other ways to do the question, such as counting all paths, and all paths through the gap, but all attempts using a b=c+d+e argument seemed quick and successful.

A couple of other tips to bear in mind on these sorts of questions.

  • When you’re deep in a calculation, it’s easy to forget that you started with a problem of some kind, and you should always go back to check that your answer makes sense in the context of the problem! Even if fits the algebra, you should not have any of the following (among many other examples that can emerge after a small conceptual or calculation error):
    • A negative radius for a circle
    • A non-integer number of configurations
    • A probability >1
  • You should also check for potential symmetries. While it’s dangerous to use symmetry excessively until you’re sure the situation genuinely is symmetric, it can be your friend in reducing errors. For example, maybe you found the final count in the far corner for Ghastly the ghost to be eg 18+16+12? But the three adjacent-to-far-corner cells are clearly equivalent from the starting point. Any difference between them can only be a feature of the argument you are trying. So they should have the same path count, and probably you made a small error in one of them.
  • It’s also worth checking your answers feel roughly right for magnitude. In particular, again on Q3 , if you get a very large answer for part b), you should try to check against other options. How many total paths are there? (Ie, ignoring the gap.) If your answer is much bigger than this, you must have made a small mistake, probably multiplying where you should have been adding. It might well, for example, not feel right if you end up showing that the number of paths is greater than the number of atoms in the universe. This applies equally well for Q4. Note that 67! is an extremely large number, and is larger than 2^100, which is the total number of colourings without observing any of the consecutive rules, let alone the maximum condition.
  • As a final point, it was good to see that many candidates were making a bona fide effort to get as many marks as possible. I would say, however, that just writing down a speculative answer is often less helpful than a single sentence explaining what you are trying to do. Pretty much any comment on patterns and a division into cases for Q4 would be useful, even if it wasn’t completely correct, whereas a guess at eg \binom{67}{33} or similar for the final answer is probably less valuable, and draws attention to what you haven’t done, rather than any progress that you really have made.

Anyway, I hope that was interesting at some level to some girls potentially interested in taking MOG or any other challenging mathematics in the future.

EGMO 2018

Last week the UK held its annual training and selection camp in Cambridge. This week, four of the students have been attending the European Girls’ Mathematical Olympiad. 2018 is the seventh edition of this prestigious competition, and is being held in Florence.

You can find very many details about the competition, and observe the UK’s excellent performance (with particular congratulations to Emily, who obtained a perfect score) at the competition website. A short article about the team in the TES can be found here.

In this post, I’m going to muse on some of the problems. You can find the two papers here and here.

Problem Two

Part a) would have been more immediate if the set A had been defined as

A:= \left\{\frac{k+1}{k} \,:\, k=1,2,3,\ldots\right\},

as this is instantly suggestive of a telescoping product such as

7 = \frac{7}{6}\cdot \frac{6}{5}\cdot\ldots \cdot \frac{2}{1}.

I found part b) to be one of the most difficult sections of the paper. It builds on the idea that given an expression for x as a product of elements of A, and an expression for y as a product of elements of A, we can combine these (ie take the product of the products!) to get an expression for xy as a product of elements of A. This yields f(xy)\le f(x)+f(y), and so the task is to show that sometimes this isn’t the most efficient way to proceed.

I tried a couple of things, like trying to bound f(p) below when p is a prime. This wasn’t ludicrous, as one would certainly need to use a term \frac{kp}{kp-1} somewhere in the product so that it is divisible by p. However, this didn’t go anywhere, and nor did investigating f(n!). Perhaps others had more success down these avenues.

But as a general rule, if an abstractly-defined function is typically hard to calculate, then classes where you can calculate it are likely to be extra valuable. Here, powers of two make life particularly easy. We have 2\in A, and so 2^n=2\times 2\times\ldots\times 2 is a valid product. And we can’t possibly achieve 2^n as a product of fewer terms than this, because 2 is the largest element of A. So f(2^n)=n. Note that this is already way better than the bound we would have achieved from our argument in part a), which yields f(m)\le m-1.

My next observation was that a similar argument and a natural construction gives f(2^n+1)=n+1. But this can be extended so that when 2^n+1\le m\le 2^{n+1}, we have f(m)\ge n+1 and in fact there is equality exactly for

m= 2^n+1, 2^n+2, 2^n+4,\ldots, 2^n+2^{n-1},2^{n+1}. (*)

In particular, note that all of these are even except 2^n+1. It may well be the case that we don’t need this extension, but before you’ve solved the problem you don’t know how much you’ll have to extend each idea!

I had a feeling that this meant f(2^n+1) was a strong candidate to satisfy the requirements, and that almost any factorisation would work. I suspect many students at this point did some refinement of choice of n, but I want to stay abstract, and use the extended observation (*). Since 2^n+1 is certainly not always prime, let’s focus on the infinitely many values n where it has a factorisation as 2^n+1 = ab, and consider whether a or b can achieve equality at (*). We’d better introduce the notation

2^\alpha<a<2^{\alpha+1},\quad 2^\beta<b<2^{\beta+1}.

So ab> 2^{ab}+2^a+2^b+1, and so \alpha+\beta>n. But similarly, ab< 2^{\alpha+1}2^{\beta+1}, so \alpha+\beta<n+2. We obtain

\alpha+\beta+1=n,

which is kind of what I’d been hoping for when I started this analysis. Now, we have

f(a)\ge \alpha+1,\quad f(b)\ge \beta+1,

\Rightarrow\quad f(a)+f(b)\ge \alpha+\beta+2 = n+1, (**)

with equality precisely if a,b both satisfy the equality conditions at (*). But a,b are odd, and so we have equality at (**) precisely if a=2^\alpha+1,b=2^\beta+1. So we have a resolution to the problem whenever 2^n+1 can be non-trivially factorised in any way other than 2^n+1=(2^\alpha+1)(2^\beta+1), so we have a rich (and certainly infinite) class of suitable (x,y).

Problem Three

An obvious remark. The jury will never choose contestant i if she has fewer than contestants in front of her unless they are forced to. They are only forced to if everyone has this property. So we ignore the second dashed bullet point, as it just tells us when the process ends. And with a little further thought, the process ends exactly when the contestants are in correct order.

I suspect part a) of this may end up featuring on future examples of our interactive write-up clinic, where students are challenged to produce technically-correct arguments for obvious but awkward mini-problems. The location of contestant C_n is probably a good starting point.

For part b), you have to find an optimal construction, and prove that it’s optimal. At national and junior olympiad level, students often forget that they have to supply both of these components. At international level, the challenge is to work out which one gives the entry into the problem. I would say that about two-thirds of the time, either the optimal construction is very obvious, or is best attacked after you’ve had some insight into the bound. For this question (and again it’s just my opinion), I felt it was all about the construction. I made absolutely no progress by aiming for bounds. Whereas the construction offers plenty of insight into how to prove the bounds, and so once I had it, I found it quick.

The usual rules apply. An optimal construction is going to have to be neat to describe. It’s very unlikely to have millions of cases. Intuitively, it seems reasonable that starting the contestants in reverse order gives the jury the greatest possible ‘elbow room’ to squeeze moves into the procedure. Perhaps you tried to prove this directly, by coupling a procedure starting from arbitrary order with a corresponding procedure starting from reverse order? Well, I found that very hard, and perhaps you did too.

However, that doesn’t mean it’s the wrong construction! The key question is, what to do about contestant C_n? Well, essentially nothing. This contestant can never be moved. So when do we start allowing other contestants to pass her? It seemed natural to let the other contestants C_1,\ldots,C_{n-1} do as much as possible among themselves first. That is

\mathbf{C_n},C_{n-1},\ldots,C_2,C_1 \quad \Rightarrow\Rightarrow\Rightarrow \quad \mathbf{C_n}, C_1,C_2,\ldots,C_{n-1},

where \Rightarrow\Rightarrow\Rightarrow denotes lots of moves. At this point, what to do next stood out for me, namely that one could use \mathbf{C_n} at the start to put all the others back into reverse order, while moving \mathbf{C_n} to the back. That is

\mathbf{C_n},C_1,C_2,\ldots,C_{n-1}\quad\Rightarrow \quad C_1,\mathbf{C_n},C_2,\ldots,C_{n-1} \quad\Rightarrow\quad C_2,C_1,\mathbf{C_n},C_3,\ldots,C_{n-1}

\Rightarrow\Rightarrow \quad C_{n-1},C_{n-2},\ldots,C_2,C_1,\mathbf{C_n}.

You might have tried other things first, but once you notice this, you just know it has to be right. It’s just too elegant a construction, and it looks like the sort of thing one prove will be optimal, because the overall process

\mathbf{C_n},C_{n-1},\ldots,C_n\quad \Rightarrow\Rightarrow\Rightarrow \quad \mathbf{C_n},C_1,C_2,\ldots,C_{n-1}

\Rightarrow\Rightarrow\quad C_{n-1},\ldots,C_2,C_1,\mathbf{C_n}\quad\Rightarrow\Rightarrow\Rightarrow\quad C_1,C_2,\ldots,C_{n-1},\mathbf{C_n},

incorporates the corresponding process for n-1 (twice, in fact) and so induction is very accessible both for calculating the total number of moves. We conjecture that this is indeed the optimal process, and under this assumption, with f(n) the number of moves, we would have f(n) = f(n-1) + (n-1) + f(n-1), from the three stages of the process, from which, after looking at small values,

f(n)=2^n - (n+1).

I started by saying that the construction was the hard part of this problem. Well, that doesn’t mean the bound is easy. But at least with a construction in hand, you can make observations that might inform a bounding argument:

  • observation 1: contestant C_n never jumps;
  • observation 2: in the optimal construction, by induction C_{n-1} doesn’t jump in the outer phases, so in fact jumps only once, ie during the middle phase;
  • observation 3: contestant C_{n-2} doesn’t jump very often, and in fact can only jump after at least one of C_{n-1} and C_n have ended up in front of her. Since we’ve established that C_{n-1},C_n don’t jump very often themselves, this gives a bound on the number of times C_{n-2}.

There is still work to do here, and errors with \pm 1 could easily creep in. But I still hold fast to my original claim that the construction was the hard part here. Or at least, the rough form of the construction. I guess it’s possible that one would have started making observations like the three above without a construction in mind, but I think it’s unlikely. Anyway, I leave for you the final details of the bounding argument, which involves formally transcribing observation 3, proving it, then generalising it to jumps of C_{n-3} and so on.

Problem Four

One of the exercises I have been setting to UK students recently is to produce short solution digests, which manifest any key ideas of the solution abstractly and briefly enough to resonate in the future. I’m a little tired of domino tiling problems, so I’ll do one of these here. This will be slightly longer than if I were not writing for a (small) audience.

Double-counting the total value by rows/columns and by dominos shows there are \frac{2kn}{3} dominos in a balanced configuration. When n=3, we can achieve k=1, and by tiling copies of this down the main diagonal, can extend to 3\,|\,n. For 3\not|\,n, we must have 3\,|\,k ie k\ge 3, but in fact k=3 is achievable, by tiling the main diagonal with copies of small boards for which k=3 can be constructed with a bit of trial-and-error.

The double-counting idea at the start is the nice part of the problem. The construction is a bit annoying, but saving ourselves work by building up large examples from copies of small examples is a useful motif to have in mind.

Problem Six

This question has lots of clues in the statement. It would, for example, be pretty surprising if the answer were ‘no’ to part b) given the setup in part a).

My confession is that I wasted lots of time on part a) not using the option m=0, which was foolish given that it’s clued from part b) that one needs to use the option m=0. My thought had been to consider some integer y, and ask which integers x were banned (if we were aiming for contradiction in part a)). For part a), it gets harder if t is smaller, so I found it helpful to think of t as \epsilon\ll 1. Anyway, if you struggled on part a), maybe worth reviewing whether you were definitely trying to solve part a), and not accidentally using the setup that really addressed part b)!

Some people have shown me solutions to part a) that carry an air of magic, by placing all the key steps (such as (*) to follow) in the language of the original setup. Let’s try to be cleaner. The key is to consider m=0. Since m=0 is included, we know that whenever x<y, we must have

\epsilon y \le x \le (1-\epsilon)y. (*)

Maybe you just have a gut feeling that this can’t be possible if you have enough xs and ys? But either way, choosing to focus on (*) is the key step, because once you know you have to prove the result based on this, it’s not too hard. I prefer addition to multiplication, so one might as well take logs (since does it really look like we’re going to use heavily the integer property now?) to obtain

\alpha\le |z_i - z_j|\le \beta,

for all z_i,z_j in some large finite collection, where 0<\alpha<\beta. You should now have a strong gut feeling that this is impossible. You have an arbitrarily large collection of real numbers which have to be close to each other pairwise, but never too close pairwise. How to finish the argument is a matter of taste.

For part b), assuming we’re aiming for the answer ‘yes’, we probably want to construct it one step at a time, and you want to think of t\approx \frac12 to make life as hard as possible.

Now, suppose you have x_1,x_2,\ldots,x_n so far. What next? Well if we could have

x_{n+1} \equiv \frac{x_i}{2}\,\mod x_i,

for all i=1,\ldots,n, that would be perfect. We can often solve sets of coupled residue equations like this using the Chinese Remainder Theorem. (Recall of course that the solutions themselves are rarely important – the fact that they exist is enough!) A couple of things go wrong with trying this crudely:

  • If x_i is odd, then \frac{x_i}{2} is not an integer…
  • If we correct this by asking for x_{n+1}\equiv \lfloor\frac{x_i}{2}\rfloor\,\mod x_i, then there’s a chance we might still be within the t-window around a multiple of x_i.
  • Unless we are going to make complicated demands on the residues, to use CRT it would be easier if all the x_is were coprime.

One option is to give up. But actually all these objections can be handled with fairly small alterations. Can you see how the second objection can be overcome by an appropriate choice of x_1? Remember that t is fixed right at the start, and cannot be equal to 1/2. Is the third objection actually an objection any more? If it is, can we fix it?

Anyway, I guess P6 was my favourite non-geometry question on the paper, though, that’s far from relevant. P5 was pretty neat too, but who knows whether a follow-up geometry post will materialise soon.

BMO2 2018

The second round of the British Mathematical Olympiad was taken yesterday by the 100 or so top scoring eligible participants from the first round, as well as some open entries. Qualifying for BMO2 is worth celebrating in its own right. The goal of the setters is to find the sweet spot of difficult but stimulating for the eligible participants, which ultimately means it’s likely to be the most challenging exam many of the candidates sit while in high school, at least in mathematics.

I know that lots of students view BMO2 as something actively worth preparing for. As with everything, this is a good attitude in moderation. Part of the goal in writing about the questions at such length (and in particular not just presenting direct solutions) is because I think at this level it’s particularly easy to devote more time than needed to preparation, and use it poorly.

All these questions could be solved by able children. In fact, each could be solved by able children in less than an hour. You definitely count as an able child if you qualified or if your teacher allowed you to make an open entry! Others count too naturally. But most candidates won’t in fact solve all the questions, and many won’t solve any. And I think candidates often come up with the wrong reasons why they didn’t solve problems. “I didn’t know the right theorems” is very very rarely the reason. Olympiad problems have standard themes and recurring tropes, but the task is not to look at the problem and decide that it is an example of Olympiad technique #371. The task is actually to have as many ideas as possible, and eliminate the ones that don’t work as quickly as possible.

The best way to realise that an idea works is to solve the problem immediately. For the majority of occasions when we’re not lucky enough for that to happen, the second-best way to realise that an idea works is to see that it makes the problem look a bit more like something familiar. Conversely, the best way to realise that an idea doesn’t work is to observe that if it worked it would solve a stronger but false problem too. (Eg Fermat’s Last Theorem *does* have solutions over the reals…) The second-best way to realise that an idea doesn’t work is to have the confidence that you’ve tried it enough and you’ve only made the problem harder, or less familiar.

Both of these second-best ideas do require a bit of experience, but I will try to explain why none of the ideas I needed for various solutions this year required any knowledge beyond the school syllabus, some similarities to recent BMOs, and a small bit of creativity.

As usual, the caveat that these are not really solutions, and certainly not official solutions, but they are close enough to spoil the problems for anyone who hasn’t tried them by themselves already. Of course, the copyright for the problems is held by BMOS, and reproduced here with permission.

Question One

I wrote this question. Perhaps as a focal point of the renaissance of my interest in geometry, or at least my interest in teaching geometry, I have quite a lot to say about the problem, its solutions, its origin story, the use of directed angles, the non-use of coordinate methods and so on. In an ideal world I would write a book about this sort of thing, but for now, a long and separate post is the answer.

This will be available once I’ve successfully de-flooded my apartment.

Question Two

I also wrote this problem, though I feel it’s only fair to show the version I submitted to the BMO committee. All the credit for the magical statement that appears above lies with them. There is a less magical origin story as well, but hopefully with some interesting combinatorial probability, which is postponed until the end of this post.One quick observation is that in my version Joe / Hatter gets to keep going forever. As we shall see, all the business happens in the first N steps, but a priori one doesn’t know that, and in my version it forces you to strategise slightly differently for Neel / Alice. In the competition version, we know Alice is done as soon as she visits a place for a second time, but not in the original. So in the original we only have to consider ‘avoid one place’ rather than the multiple possibilities now of ‘avoid one place’ or ‘visit a place again’.

But I think the best idea is to get Alice to avoid one particular place c\not\equiv 0 whenever possible. At all times she has two possible options for where to go next, lets say b_k+a_k, b_k-a_k in the language of the original statement. We lose nothing by assuming -N/2 < a_k\le N/2, and certainly it would be ridiculous for Joe / Hatter ever to choose a_k=0. The only time Alice’s strategy doesn’t work is when both of these are congruent to c, which implies N\,|\, 2a_k, and thus we must have N= 2a_k. In other words, Alice’s strategy will always work if N is odd.

I think it’s really worth noticing that the previous argument is weak. We certainly did not show that N must be odd for Alice to win. We showed that Alice can avoid a congruence class modulo an odd integer. We didn’t really need that odd integer to be N for this to work. In particular, if N has an odd factor p (say a prime), then the same argument works to show that we can avoid visiting any site with label congruent to 1 modulo p.

It’s actually very slightly more complicated. In the original argument, we didn’t need to use any property of b_k. But obviously here, if b_k\equiv 1 modulo p and p\,|\,a_k, then certainly b_{k+1}\equiv 1 modulo p. So we have to prove instead that Alice can ensure she never ‘visits 1 modulo p for the first time’. Which is fine, by the same argument.

So, we’ve shown that Neel / Alice wins if N is odd, or has an odd factor. The only values that remain are powers of 2. I should confess that I was genuinely a little surprised that Joe / Hatter wins in the power of 2 case. You can find a construction fairly easily for N=2 and N=4, but I suspected that might be a facet of small numbers. Why? Because it still felt we could avoid a particular site. In order for Alice’s strategy to fail, we have to end up exactly opposite the particular site at exactly the time when the next a_k=N/2, and so maybe we could try to avoid that second site as well, and so on backwards?

But that turned out to be a good example of something that got very complicated quite quickly with little insight. And, as discussed at the beginning, that’s often a sign in a competition problem that your idea isn’t so good. (Obviously, when composing a problem, that’s no guarantee at all. Sometimes things are true but no good ideas work.) So we want other ideas. Note that for N=4, the sequence (2,1,2) works for Joe / Hatter, because that forces Alice / Neel to visit either (0,2,1,3) or (0,2,3,1). In particular, this strategy gave Alice no control on the first step nor the last step, and the consequence is that we force her to visit the evens first, then transfer to an odd, and then force her to visit the other odd.

We might play around with N=8, or we might proceed directly to a general extension. If we have a Joe / Hatter strategy for N, then by doubling all the a_ks, we have a strategy for 2N which visits all the even sites in the first N steps. But then we can move to an odd site eg by taking a_N=1. Just as in the N=4 case, it doesn’t matter which odd site we start from, since if we again double all the a_ks, we will visit all the other odd sites. This gives us an inductive construction of a strategy for powers of two. To check it’s understood, the sequence for N=8 is (4,2,4,1,4,2,4).

Although we don’t use it, note that this strategy takes Alice on a tour of sites described by decreasing order of largest power of two dividing the label of the site.

Continue reading

BMO1 2017 – Questions 5 and 6

The first round of the British Mathematical Olympiad was sat yesterday. The questions can be found here and video solutions here. My comments on the first four questions are in the previous post.

Overall, I didn’t think any of the questions on this paper were unusually difficult by the standard of BMO1, but I found everything slightly more time-consuming than typical. I thought Question 5 was a great problem, and I tried lots of things unsuccessfully, first, and so wanted to discuss it in slightly more technical language. For Question 6 I made a decisive mistake, which I’ll explain, and which cost a lot of time. But in general, my point is that the back end of the paper was a little fiddlier than normal, and required longer written solutions, and perhaps many students might have had less time than expected to attack them anyway after details earlier in the paper.

Question Five

As I said before, I thought this question was quite challenging. Not because the solution is particularly exotic or complicated, but because there were so many possible things that might have worked. In my opinion it would not have been out of place at the start of an IMO paper, because it’s perfectly possible to have enough good ideas that eliminating the ones that don’t work takes an hour, or hours. Even though it slightly spoils the flow of the solution, I’m particularly trying to emphasise the tangents that didn’t work, mostly for reassurance to anyone who spent a long time struggling.

I was thinking about this question in terms of a 2Nx2N board, where N is even, and for the given question equal to 100. I spent a while thinking that the bound was 8N-4, corresponding to taking the middle two rows and the middle two columns, but not the 2×2 square which is their intersection. If you think of a comb as a ‘handle’ of 1xN cells, with an extra N/2 alternating cells (say, ‘teeth’) bolted on, then it’s clear this construction works because there’s never space to fit in a handle, let alone the teeth.

I couldn’t prove that this was optimal though. A standard way to prove a given bound K was optimal would be to produce a tiling on the board with K combs, where every cell is included in exactly one comb. But this is clearly not possible in this situation, since the number of cells in a comb (which is 150) does not divide the total number of cells on the board.

Indeed, the general observation that if you take a comb and a copy of the comb rotated by 180, the teeth of the second comb can mesh perfectly with the teeth of the first comb to generate a 3xN unit. I wasted a moderate amount of time pursuing this route.

[Note, it will be obvious in a minute why I’m writing ‘shaded’ instead of ‘coloured’.]

But in motivating the construction, I was merely trying to shade cells so that they intersected every possible 1xN handle, and maybe I could prove that it was optimal for this. In fact, I can’t prove it’s optimal because it isn’t optimal – indeed it’s clear that a handle through one of the middle rows intersects plenty of shaded cells, not just one. However, with this smaller problem in mind, it didn’t take long to come up with an alternative proposal, namely splitting the board into equal quarters, and shading the diagonals of each quarter, as shown.

It seems clear that you can’t fit in a 1xN handle, and any sensible tiling with 1xN handles contains exactly one shaded cell, so this shading (with 4N shaded cells) is optimal. But is it optimal for a comb itself?

Consider a shading which works, so that all combs include a shaded cell. It’s clear that a comb is contained within a 2xN block, and in such a 2xN block, there are four possible combs, as shown.

You need to cover all these combs with some shading somewhere. But if you put the shaded cell on a tooth of comb A, then you haven’t covered comb B. And if you put the shaded cell on the handle of comb A, then you haven’t covered one of comb C and comb D. You can phrase this via a colouring argument too. If you use four colours with period 2×2, as shown

then any comb involves exactly three colours, and so one of them misses out the colour of the shaded cell. (I hope it’s clear what I mean, even with the confusing distinction between ‘shaded’ and ‘coloured’ cells.)

Certainly we have shown that any 2xN block must include at least two shaded cells. And that’s pretty much it. We have a tiling with 2N copies of a 2xN block, and with at least two shaded cells in each, that adds to at least 4N shaded cells overall.

Looking back on the method, we can identify another way to waste time. Tiling a board, eg a chessboard with dominos is a classic motif, which often relies on clever colouring. So it’s perhaps lucky that I didn’t spot this colouring observation earlier. Because the argument described really does use the local properties of how the combs denoted A-D overlap. An attempt at a global argument might start as follows: we can identify 2N combs which don’t use colour 1, and tile this subset of the grid with them, so we need to shade at least 2N cells from colours {2,3,4}. Similarly for sets of colours {1,3,4}, {1,2,4}, and {1,2,3}. But if we reduce the problem to this, then using roughly 2N/3 of each colour fits this global requirement, leading to a bound of 8N/3, which isn’t strong enough. [1]

Question Six

A word of warning. Sometimes it’s useful to generalise in problems. In Q5, I was thinking in terms of N, and the only property of N I used was that it’s even. In Q4, we ignored 2017 and came back to it at the end, using only the fact that it’s odd. By contrast, in Q2, the values did turn out to be important for matching the proof bounds with a construction.

You have to guess whether 300 is important or not here. Let’s see.

I have a natural first question to ask myself about the setup, but some notation is useful. Let a_1,a_2,\ldots,a_{300} be the ordering of the cards. We require that \frac{a_1+\ldots+a_n}{n} is an integer for every 1\le n\le 300. Maybe the values of these integers will be important, so hold that thought, but for now, replace with the divisibility statement that n | a_1+\ldots+a_n.

I don’t think it’s worth playing with small examples until I have a better idea whether the answer is 5 or 295. So the natural first question is: “what does it mean to have (a_1,\ldots,a_{n-1}) such that you can’t pick a suitable a_n?”

It means that there is no integer k in \{1,\ldots,300\}\backslash\{a_1,\ldots,a_{n-1}\} such that n\,\big|\,(a_1+\ldots+a_{n-1})+k, which for now we write as

k\equiv -(a_1+\ldots+a_{n-1})\,\mod n.

Consider the congruence class of -(a_1+\ldots+a_{n-1}) modulo n. There are either \lfloor \frac{300}{n}\rfloor or \lceil \frac{300}{n}\rceil integers under consideration in this congruence class. If no such k exists, then all of the relevant integers in this congruence class must appear amongst \{a_1,\ldots,a_{n-1}\}. At this stage, we’re trying to get a feel for when this could happen, so lower bounds on n are most relevant. Therefore, if we get stuck when trying to find a_n, we have

\lfloor \frac{300}{n} \rfloor\text{ or }\lceil \frac{300}{n}\rceil \le n-1, (*)

which is summarised more succinctly as

\lfloor \frac{300}{n} \rfloor \le n-1. (**)

[Note, with this sort of bounding argument, I find it helpful to add intermediate steps like (*) in rough. The chance of getting the wrong direction, or the wrong choice of \pm 1 is quite high here. Of course, you don’t need to include the middle step in a final write-up.]

We can check that (**) is false when n\le 17 and true when n\ge 18. Indeed, both versions of (*) are true when n\ge 18.

So we know the minimum failure length is at least 17. But is there a failing sequence of length 17? At a meta-level, it feels like there should be. That was a very natural bounding argument for 17 (which recall corresponds to n=18), and it’s easy to believe that might be part of an official solution. If we achieve equality throughout the argument, that’s most of the way to a construction as well. It won’t be so easy to turn this argument into a construction for n\ge 19 because there won’t be equality anywhere.

We have to hope there is a construction for n=18. What follows is a description of a process to derive (or fail to derive) such a construction. In a solution, one would not need to give this backstory.

Anyway, in such a construction, let \alpha\in\{1,2,\ldots,18\} describe the congruence class modulo 18 which is exhausted by \{a_1,\ldots,a_{17}\}. I’m going to hope that \alpha=18 because then the calculations will be easier since everything’s a multiple of 18. We haven’t yet used the fact that for a problem, we need \alpha\equiv-(a_1+\ldots+a_{n-1}). We definitely have to use that. There are 16 multiples of 18 (ie relevant integers in the congruence class), so exactly one of the terms so far, say a_j, is not a multiple of 18. But then

0 \equiv 0+\ldots+0+a_j+0+\ldots+0,

which can’t happen. With a bit of experimentation, we find a similar problem making a construction using the other congruence classes with 16 elements, namely \alpha\in \{13,14,\ldots,18\}.

So we have to tackle a different class. If \alpha\le 12 then our sequence must be

\alpha,18+\alpha,2\times 18 +\alpha, \ldots, 16\times 18 + \alpha,

in some order. In fact, let’s add extra notation, so our sequence is

(a_1,\ldots,a_{17}) = (18\lambda_1+ \alpha,\ldots,18\lambda_{17}+\alpha),

where (\lambda_1,\ldots,\lambda_{17}) is a permutation of {0,…,16}. And so we require

k \,\big|\, 18(\lambda_1+\ldots+\lambda_k) + k\alpha, (%)

for 1\le k\le 17. But clearly we can lop off that k\alpha, and could ignore the 18. Can we find a permutation \lambda such that

k \,\big|\, \lambda_1+\ldots+\lambda_k.

This was where I wasted a long time. I played around with lots of examples and kept getting stuck. Building it up one term at a time, I would typically get stuck around k=9,10. And I had some observations that in all the attempted constructions, the values of \frac{\lambda_1+\ldots+\lambda_k}{k} were around 8 and 9 too when I got stuck.

I became convinced this subproblem wasn’t possible, and decided that would be enough to show that n=18 wasn’t a possible failure length. I was trying to show the subproblem via a parity argument (how must the a_is alternate odd/even to ensure all the even partial sums are even) but this wasn’t a problem. Then I came up with a valid argument. We must have

\lambda_1+\ldots+\lambda_{17}=136= 16\times 8 + 8\quad\text{and}\quad 16\,\big|\,\lambda_1+\ldots+\lambda_{16},

which means \lambda_1+\ldots+\lambda_{16} must be 128 = 15×8 + 8, ie \lambda_{17}=8. But then we also have 15\,\big|\, \lambda_1+\ldots+\lambda_{15}, which forces $latex\lambda_{16}=8$ also. Which isn’t possible.

If this then hadn’t wasted enough time, I then tried to come up with a construction for n=19, for which there are lots more variables, and took a lot more time, and seemed to be suffering from similar problems, just in a more complicated way. So I became convinced I must have made a mistake, because I was forced down routes that were way too complicated for a 3.5 hour exam. Then I found it…

What did I do wrong? I’ll just say directly. I threw away the 18 after (%). This made the statement stronger. (And in fact false.) Suppose instead I’d thrown away a factor of 9 (or no factors at all, but it’s the residual 2 that’s important). Then I would be trying to solve

k\,\big|\,2(\lambda_1+\ldots+\lambda_k).

And now if you experiment, you will notice that taking \lambda_1=0,\lambda_2=1,\lambda_3=2,\ldots seems to work fine. And of course, we can confirm this, using the triangle number formula for the second time in the paper!

This had wasted a lot of time, but once that thought is present, we’re done, because we can go straight back and exhibit the sequence

(a_1,\ldots,a_{17}) = (1, 18+1,2\times 18 +1,\ldots, 16\times 18 +1).

Then the sum so far is congruent to -1 modulo 18, but we have exhausted all the available integers which would allow the sum of the first 18 terms to be a multiple of 18. This confirms that the answer to the question as stated is 17.

At the start, I said that we should be cautious about generalising. In the end, this was wise advice. We definitely used the fact that 18 was even in the stage I over-reduced the first time. We also used the fact that there was at least one value of \alpha with an ‘extra’ member of the congruence class. So I’m pretty sure this proof wouldn’t have worked with 288 = 16×18 cards.

Footnotes

[1] – If shading were a weighted (or continuous or whatever you prefer) property, ie that each cell has a quantity of shading given by a non-negative real number, and we merely demand that the total shading per comb is at least one, then the bound 8N/3 is in fact correct for the total shading. We could look at a 2xN block, and give 1/3 shading to one cell of each colour in the block. Alternatively, we could be very straightforward and apply 2/3N shading to every cell in the grid. The fact that shading has to be (in this language) zero or one, imposes meaningful extra constraints which involve the shape of the comb.

BMO1 2017 – Questions 1-4

The first round of the British Mathematical Olympiad was sat yesterday. The questions can be found here. I recorded some thoughts on the questions while I was in Cyprus, hence the nice Mediterranean sunset above. I hope this might be useful to current or future contestants, as a supplement to the concise official solutions available. It goes without saying that while these commentaries may be interesting at a general level, they will be much more educational to students who have at least digested and played around with the questions, so consider trying the paper first. Video solutions are available here. These have more in common with this blog post than the official solutions, though inevitably some of the methods are slightly different, and the written word has some merits and demerits over the spoken word for clarity and brevity.

The copyright for these questions lies with BMOS, and are reproduced here with permission. Any errors or omissions are obviously my own.

I found the paper overall quite a bit harder than in recent years, or at least harder to finish quickly. I’ve therefore postponed discussion of the final two problems to a second post, to follow shortly.

Question One

A recurring theme of Q1 from BMO1 in recent years has been: “it’s possible to do this problem by a long, and extremely careful direct calculation, but additional insight into the setup makes life substantially easier.”

This is the best example yet. It really is possible to evaluate Helen’s sum and Phil’s sum, and compare them directly. But it’s easy to make a mistake in recording all the remainders when the divisor is small, and it’s easy to make a mistake in summation when the divisor is large, and so it really is better to have a think for alternative approaches. Making a mistake in a very calculation-heavy approach is generally penalised heavily. And this makes sense intellectually, since the only way for someone to fix an erroneous calculation is to repeat it themselves, whereas small conceptual or calculation errors in a less onerous solution are more easily isolated and fixed by a reader. Of course, it also makes sense to discourage such attempts, which aren’t really related to enriching mathematics, which is the whole point of the exercise!

Considering small divisors (or even smaller versions of 365 and 366) is sometimes helpful, but here I think a ‘typical’ divisor is more useful. But first, some notation will make any informal observation much easier to turn into a formal statement. Corresponding to Helen and Phil, let h(n) be the remainder when n is divided by 365, and p(n) the remainder when n is divided by 366. I would urge students to avoid the use of ‘mod’ in this question, partly because working modulo many different bases is annoying notationally, partly because the sum is not taken modulo anything, and partly because the temptation to use mod incorrectly as an operator is huge here [1].

Anyway, a typical value might be n=68, and we observe that 68 x 5 + 25 = 365, and so h(68)=25 and p(68)=26. Indeed, for most values of n, we will have p(n)=h(n)+1. This is useful because

p(1)+p(2)+\ldots+p(366) - \left(h(1)+h(2)+\ldots+h(365)\right)

= \left(p(1)-h(1)\right) + \ldots+\left(p(365)-h(365)\right) + p(366),

and now we know that most of the bracketed terms are equal to one. We just need to handle the rest. The only time it doesn’t hold that p(n)=h(n)+1 is when 366 is actually a multiple of n. In this case, p(n)=0 and h(n)=n-1. We know that 366 = 2 x 3 x 61, and so its divisors are 1, 2, 3, 6, 61, 122, 183.

Then, in the big expression above, seven of the 365 bracketed terms are not equal to 1. So 358 of them are equal to one. The remaining ones are equal to 0, -1, -2, -5, -60, -121, -182 respectively. There are shortcuts to calculate the sum of these, but it’s probably safer to do it by hand, obtaining -371. Overall, since p(366)=0, we have

p(1)+p(2)+\ldots+p(366) - \left(h(1)+h(2)+\ldots+h(365)\right)

= -371 + 358 + 0 = -13.

So, possibly counter-intuitively, Helen has the larger sum, with difference 13, and we didn’t have to do a giant calculation…

Question Two

Suppose each person chooses which days to go swimming ‘at random’, without worrying about how to define this. Is this likely to generate a maximum or minimum value of n? I hope it’s intuitively clear that this probably won’t generate an extreme value. By picking at random we are throwing away lots of opportunity to force valuable overlaps or non-overlaps. In other words, we should start thinking about ways to set up the swimming itinerary with lots of symmetry and structure, and probably we’ll eventually get a maximum or a minimum. At a more general level, with a problem like this, one can start playing around with proof methods immediately, or one can start by constructing lots of symmetric and extreme-looking examples, and see what happens. I favour the latter approach, at least initially. You have to trust that at least one of the extreme examples will be guess-able.

The most obvious extreme example is that everyone swims on the first 75 days, and no-one swims on the final 25 days. This leads to n=75. But we’re clearly ‘wasting’ opportunities in both directions, because there are never exactly five people swimming. I tried a few more things, and found myself simultaneously attacking maximum and minimum, which is clearly bad, so focused on minimum. Just as a starting point, let’s aim for something small, say n=4. The obstacle is that if you demand at most four swimmers on 96 days, then even with six swimmers on the remaining four days, you don’t end up with enough swimming having taken place!

Maybe you move straight from this observation to a proof, or maybe you move straight to a construction. Either way, I think it’s worth saying that the proof and the construction come together. My construction is that everyone swims on the first 25 days, then on days 26-50 everyone except A and B swim, on days 51-75 everyone except C and D swim, and on days 76-100 everyone except E and F swim. This exactly adds up. And if you went for the proof first, you might have argued that the total number of swim days is 6×75 = 450, but is at most 4n + 6(100-n). This leads immediately to n\ge 25, and I just gave the construction. Note that if you came from this proof first, you can find the construction because your proof shows that to be exact you need 25 days with six swimmers, and 75 days with four swimmers, and it’s natural to try to make this split evenly. Anyway, this clears up the minimum.

[Less experienced contestants might wonder why I was worried about generating a construction despite having a proof. Remember we are trying to find the minimum. I could equally have a proof for n\ge 10 which would be totally totally valid. But this wouldn’t show that the minimum was n=10, because that isn’t in fact possible (as we’ve seen), hence it’s the construction that confirms that n=25 is the true minimum.]

It’s tempting to go back to the drawing board for the maximum, but it’s always worth checking whether you can directly adjust the proof you’ve already given. And here you can! We argued that

450\le 4n + 6(100-n)

to prove the minimum. But equally, we know that on the n days we have at least five swimmers, and on the remaining days, we have between zero and four swimmers, so

450 \ge 5n + 0\times (100-n), (*)

which gives n\le 90. If we have a construction that attains this bound then we are done. Why have I phrased (*) with the slightly childish multiple of zero? Because it’s a reminder that for a construction to attain this bound, we really do need the 90 days to have exactly five swimmers, and the remaining ten days to have no swimmers. So it’s clear what to do. Split the first 90 days into five groups of 15 days. One swimmer skips each group. No-one swims in the final ten days, perhaps because of a jellyfish infestation. So we’re done, and 25\le n\le 90.

At a general level, it’s worth noting that in the story presented, we found an example for the minimum which we turned into a proof, and then a proof for the maximum, which we then analysed to produce a construction.

Note that similar bounding arguments would apply if we fiddled with the numbers 5, 75 and 100. But constructions matching the bounds might not then be possible because the splits wouldn’t work so nicely. This would make everything more complicated, but probably not more interesting.

Question Three

It’s understandable that lots of students attempting this paper might feel ill-at-ease with conventional Euclidean geometry problems. A good first rule of thumb here, as in many settings, is “don’t panic!”, and a more specific second rule of thumb is “even if you think you can calculate, try to find geometric insight first.”

Here, it really does look like you can calculate. A configuration based on a given isosceles triangle and a length condition and a perpendicular line is open to several coordinate approaches, and certainly some sensible trigonometry. It’s also very open to organised labelling of the diagram. You have three equal lengths, and a right-angle, as shown.

The key step is this. Drop the perpendicular from A to BC, and call its foot D. That alone really is the key step, as it reduces both parts of the question to an easy comparison. It’s clear that the line AD splits the triangle into two congruent parts, and thus equal areas and perimeters. So it is enough to show that triangle BMN has the same area as triangle ABD, and that their outer-perimeters (ie the part of its perimeter which is also the perimeter of ABC) are the same.

But they’re congruent, so both of these statements are true, and the problem is solved.

My solution could be as short as two or three lines, so for the purposes of this post all that remains is to justify why you might think of the key step. Here are a few possible entry routes:

  • You might notice that line AD induces the required property for triangle ABD.
  • You might try to find a triangle congruent to AMN, and come up with D that way.
  • There’s already a perpendicular in the question so experimenting with another one is natural, especially since the perpendicular from A has straightforward properties.
  • AMN is a right angle, and so constructing D gives a cyclic quadrilateral. We didn’t use that directly in the proof above, but constructing cyclic quadrilaterals is usually a good idea.
  • If you were trying a calculation approach, you probably introduced the length AD, or at least the midpoint D as an intermediate step.

On the video, Mary Teresa proposes a number of elegant synthetic solutions with a few more steps. You might find it a useful exercise to try to come up with some motivating reasons like the bullet points above to justify her suggestion to reflect A in M as a first step.

Question Four

BMO1 2017Q4

I wasn’t paying enough attention initially, and I calculated a_2=0\text{ or }2. This made life much much more complicated. As with IMO 2017 Q1, if trying to deduce general behaviour from small examples, it’s essential to calculate the small examples correctly!

Once you engage your brain properly, you find that a_2=0 \text{ or }3, and of course a_2=0 is not allowed, since it must be positive. So a_2=3, and a similar calculation suggests a_3=1\text{ or }6. It’s clear that the set of values for a_{k+1} depends only on a_k, so if you take a_3=1, then you’re back to the situation you started with at the beginning. If you choose to continue the exploration with a_3=6, you will find a_4=2\text{ or }10, at which point you must be triggered by the possibility that triangle numbers play a role here.

As so often with a play-around with small values, you need to turn a useful observation into a concrete statement, which could then be applied to the problem statement. It looks like in any legal sequence, every term will be a triangle number, so we only need to clarify which triangle number. An example of a suitable statement might be:

Claim: If a_n=T_k=\frac{k(k+1)}{2}, the k-th triangle number, then a_{n+1}=T_{k-1}\text{ or }T_{k+1}.

There are three stages. 1) Checking the claim is true; 2) checking the claim is maximally relevant; 3) proving it. In this case, proving it is the easiest bit. It’s a quick exercise, and I’m omitting it. Of course, we can’t prove any statement which isn’t true, and here we need to make some quick adjustment to account for the case k=1, for which we are forced to take a_{n+1}=T_{k+1}.

The second stage really concerns the question “but what if a_n\ne T_k?” While there are deductions one could make, the key is that if a_1 is a triangle number, the claim we’ve just made shows that a_n is always a triangle number, so this question is irrelevant. Indeed the claim further shows that a_{2017}\le T_{2017}, and also that a_{2017}=T_k for some odd value of k. To be fully rigorous you should probably describe a sequence which attains each odd value of k, but this is really an exercise in notation [2], and it’s very obvious they are all attainable.

In any case, the set of possible values is \{T_1,T_3,\ldots,T_{2017}\}, which has size 1009.

Final two questions

These are discussed in a subsequent post.

Footnotes

[1] – mod n is not an operator, meaning you shouldn’t think of it as ‘sending integers to other integers’, or ‘taking any integer, to an integer in {0,1,…,n-1}’. Statements like 19 mod 5 = 4 are useful at the very start of an introduction to modular arithmetic, but why choose 4? Sometimes it’s more useful to consider -1 instead, and we want statements like a^p\equiv a modulo p to make sense even when a\ge p. 19 = 4 modulo 5 doesn’t place any greater emphasis on the 4 than the 19. This makes it more like a conventional equals sign, which is of course appropriate.

[2] – Taking a_n=T_n for $1\le n\le k$, and thereafter a_n=T_k if k is odd, and $a_n=T_{k+1}$ if k is even will certainly work, as will many other examples, some perhaps easier to describe than this one, though make sure you don’t accidentally try to use T_0!

Characterising fixed points in geometry problems

There’s a risk that this blog is going to become entirely devoted to Euclidean geometry, but for now I’ll take that risk. I saw the following question on a recent olympiad in Germany, and I enjoyed it as a problem, and set it on a training sheet for discussion with the ten British students currently in contention for our 2017 IMO team.

Given a triangle ABC for which AB\ne AC. Prove there exists a point D\ne A on the circumcircle satisfying the following property: for any points M,N outside the circumcircle on rays AB, AC respectively, satisfying BM=CN, the circumcircle of AMN passes through D.

Proving the existence of a fixed point/line/circle which has a common property with respect to some other variable points/lines/circles is a common style of problem. There are a couple of alternative approaches, but mostly what makes this style of problem enjoyable is the challenge of characterising what the fixed point should be. Sometimes an accurate diagram will give us everything we need, but sometimes we need to be clever, and I want to discuss a few general techniques through the context of this particular question. I don’t want to make another apologia for geometry as in the previous post, but if you’re looking for the ‘aha moment’, it’ll probably come from settling on the right characterisation.

At this point, if you want to enjoy the challenge of the question yourself, don’t read on!

Reverse reconstruction via likely proof method

At some point, once we’ve characterised D in terms of ABC, we’ll have to prove it lies on the circumcircle of any AMN. What properties do we need it to have? Well certainly we need the angle relation BDC = A, but because MDAN will be cyclic too, we also need the angle relation MDN = A. After subtracting, we require angles MDB = NDC.

Depending on your configuration knowledge, this is all quite suggestive. At the very least, when you have equal angles and equal lengths, you might speculate that the corresponding triangles are congruent. Here that would imply BD=CD, which characterises D as lying on the perpendicular bisector of BC. D is also on the circumcircle, so in fact it’s also on the angle bisector of BAC, here the external angle bisector. This is a very common configuration (normally using the internal bisector) in this level of problem, and if you see this coming up without prompting, it suggests you’re doing something right.

So that’s the conjecture for D. And we came up with the conjecture based on a likely proof strategy, so to prove it, we really just need to reverse the steps of the previous two paragraphs. We now know BD=CD. We also know angles ABD = ACD, so taking the complementary angles (ie the obtuse bit in the diagram) we have angles DBM = DCN, so we indeed have congruent triangles. So we can read off angles MDB = NDC just as in our motivation, and recover that MDAN is cyclic.

Whatever other methods there are to characterise point D (to follow), all methods will probably conclude with an argument like the one in this previous paragraph, to demonstrate that D does have the required property.

Limits

We have one degree of freedom in choosing M and N. Remember that initially we don’t know what the target point D is. If we can’t see it immediately from drawing a diagram corresponding to general M and N, it’s worth checking some special cases. What special cases might be most relevant depends entirely on the given problem. The two I’m going to mention here both correspond to some limiting configuration. The second of these is probably more straightforward, and was my route to determining D. The first was proposed by one of my students.

First, we conjecture that maybe the condition that M and N lie outside the circumcircle isn’t especially important, but has been added to prevent candidates worrying about diagram dependency. The conclusion might well hold without this extra stipulation. Remember at this stage we’re still just trying to characterise D, so even if we have to break the rules to find it, this won’t damage the solution, since we won’t be including our method for finding D in our written-up solution!

Anyway, WLOG AC < AB. If we take N very close to A, then the distances BM and MA are c and b-c respectively. The circumcircle of AMN is almost tangent to line AC. At this point we stop talking about ‘very close’ and ‘almost tangent’ and just assume that N=A and the so the circle AMN really is the circle through M, tangent to AC at A. We need to establish where this intersects the circumcircle for a second time.

To be clear, I found what follows moderately tricky, and this argument took a while to find and was not my first attempt at all. First we do some straightforward angle-chasing, writing A,B,C for the measures of the angles in triangle ABC. Then the angle BDC is also A and angle BDA is 180-C. We also have the tangency relation from which the alternate segment theorem gives angle MDA = A. Then BDM = BDA – MDA = 180 – C – A = B. So we know the lengths and angles in the configuration BDAM.

At this point, I had to use trigonometry. There were a couple of more complicated options, but the following works. In triangle BDM, a length b is subtended by angle B, as is the case for the original triangle ABC. By the extended sine rule, BDM then has the same circumradius as ABC. But now the length BD is subtended by angle DMB in one of these circumcircles, and by DAB in the other. Therefore these angles are either equal or complementary (in the sense that they sum to 180). Clearly it must be the latter, from which we obtain that angles DMA = MAD = 90 – A/2. In other words, D lies on the external angle bisector of A, which is the characterisation we want.

Again to clarify, I don’t think this was a particularly easy or particularly natural argument for this exact problem, but it definitely works, and the idea of getting a circle tangent to a line as a limit when the points of intersection converge is a useful one. As ever, when an argument uses the sine rule, you can turn it into a synthetic argument with enough extra points, but of the options I can currently think of, I think this trig is the cleanest.

My original construction was this. Let M and N be very very far down the rays. This means triangle AMN is large and approximately isosceles. This means that the line joining A to the circumcentre of AMN is almost the internal angle bisector of MAN, which is, of course, also the angle bisector of BAC. Also, because triangle AMN is very large, its circumcircle looks, locally, like a line, and has to be perpendicular to the circumradius at A. In other words, the circumcircle of AMN is, near A, approximately line perpendicular to the internal angle bisector of BAC, ie the external angle bisector of BAC. My ‘aha moment’ factor on this problem was therefore quite high.

Direct arguments

A direct argument for this problem might consider a pairs of points (M,N) and (M’,N’), and show directly that the circumcircles of ABC, AMN and AM’N’ concur at a second point, ie are coaxal. It seems unlikely to me that an argument along these lines wouldn’t find involve some characterisation of the point of concurrency along the way.

Do bear in mind, however, that such an approach runs the risk of cluttering the diagram. Points M and N really weren’t very important in anything that’s happened so far, so having two pairs doesn’t add extra insight in any of the previous methods. If this would have been your first reaction, ask yourself whether it would have been as straightforward or natural to find a description of D which led to a clean argument.

Another direct argument

Finally, a really neat observation, that enables you to solve the problem without characterising D. We saw that triangles DBM and DCN were congruent, and so we can obtain one from the other by rotating around D. We say D is the centre of the spiral similarity (here in fact with homothety factor 1 ie a spiral congruence) sending BM to CN. Note that in this sort of transformation, the direction of these segments matters. A different spiral similarity sends BM to NC.

But let’s take any M,N and view D as this spiral centre. The transformation therefore maps line AB to AC and preserves lengths. So in fact we’ve characterised D without reference to M and N ! Since everything we’ve said is reversible, this means as M and N vary, the point we seek, namely D, is constant.

This is only interesting as a proof variation if we can prove that D is the spiral centre without reference to one of the earlier arguments. But we can! In general a point D is the centre of spiral similarity mapping BM to CN iff it is also the centre of spiral similarity mapping BC to MN. And we can find the latter centre of spiral similarity using properties of the configuration. A is the intersection of MB and CN, so we know precisely that the spiral centre is the second intersection point of the two circumcircles, exactly as D is defined in the question.

(However, while this is cute, it’s somehow a shame not to characterise D as part of a solution…)

Symmedians and Balkan MO 2017 Q2

While I was away, I wrote about my latest approach to teaching geometry at olympiad camps. This post will end up being about Q2 from the Balkan MO which took place yesterday in Macedonia, but first there is quite a long prelude. My solution, and probably many solutions, to this problem made use of a standard configuration in triangle geometry, namely the symmedian. I want to introduce the configuration, give some simpler examples in practice, and along the way talk about my slightly patched-together philosophy about the merits of practising Euclidean geometry.

The symmedian

Draw a triangle ABC, with A at the top of the page, and extend the rays AB and AC. The median is the line from A through M, the midpoint of BC. Now take points D and E on AB and AC respectively. The following properties are equivalent:

  • DE is parallel to BC;
  • triangle ADE is similar to triangle ABC;
  • the median of ABC passes through the midpoint of DE, and thus is also the median of ADE.

I think it’s a little awkward to prove either of the first two from the third – ratios of areas works – but the rest of the equivalences are straightforward. Later I’m going to talk about the difference between an exercise and a problem. These are all, at best, exercises.

Now take B’ on the ray AC, and C’ on the ray AB such that triangle AB’C’ is similar to triangle ABC. One way to achieve this is to take B’ and C’ to be the reflections in the angle bisector of A of B and C respectively (so then AB’=AB and AC’=AC). We say the line B’C’ is antiparallel to BC, as is any other line DE parallel to B’C’. (Probably this should say ‘with respect to triangle ABC’ or similar, but the context here is very clear, and I want this to seem natural rather than opaque.) Note that DE is an antiparallel line iff BCED is a cyclic quadrilateral. We should remember that, as cyclic quadrilaterals are the signposts for progress in both exercises and problems.

The median of triangle AB’C’ obeys the same equivalences as described above, and so bisects any antiparallel segment. We call the median of triangle AB’C’ the symmedian of triangle ABC. Using the first set of equivalences, the symmedian of triangle ABC bisects any line antiparallel to BC. Furthermore, by construction, the symmedian is the image of the median of ABC under reflection in the bisector of the angle at A. We sometimes say that the symmedian is the isogonal conjugate of the median.

That’s my definition. Note that there was essentially one definition then a couple of easy equivalent definitions. At no point again will I discuss the equivalence of these definitions – we have to take that for granted if we want to get on to more interesting things.

Intersection of tangents + concurrency

Now, in triangle ABC, draw the tangents to the circumcircle at B and C. These meet at P. It turns out that AP is the symmedian. This could have been our definition of a symmedian, but it wasn’t, so let’s quickly prove this.

Trigonometric arguments are very accessible, but I’ll give a Euclidean argument. Draw the antiparallel DE through P, as shown. Our task is to show that EP=PD. At this point, I would again say that this is an exercise.

We colour the angle ABC in green. Two angles around point C share this measure by the alternate segment theorem. The angle at E shares this measure because DE is antiparallel. Therefore CPE is isosceles, and so EP=CP. But CP=BP, so by applying the same argument for the orange angles, we get EP=CP=BP=DP as required.

Pause to regroup. Proving this wasn’t hard, but it was perhaps surprising. If this was all new to you, and I told you to consider the reflection of the median in the angle bisector, you probably wouldn’t instantly exclaim “it goes through the tangent intersection!” So this is a useful piece of knowledge to have gained, in case we ever have to work with the intersection of two tangents like this. Maybe it won’t be useful, but maybe it will. Maybe the statement itself won’t but some extra insights from the proof will be useful, like the fact that we actually showed P is the centre of the circle BCED, and thus angles ECD=EBD=90.

A second property is that in a triangle ABC, the symmedian from A, the symmedian from B and the symmedian from C intersection at, naturally, the symmedian point, which is usually denoted K. This comes from the fact that each symmedian is the isogonal conjugate of the respective median, and the medians are known to concur at the centroid. I’m not going to get into this now.

Configurations – an example

Here’s a problem. Take an isosceles trapezium ABCD as shown (ie throughout I don’t want to worry about alternative diagrams).

Let M be the midpoint of AD, and let E be the point on CM such that angle DBM = EBA. Prove that ABCDE is cyclic.

Well, certainly ABCD is cyclic. So we just need to show E also lies on that circle. And we have two equal angles, but they aren’t in the right place to conclude this immediately. However, we have angle MCA = DBM = EBA, so ABCE is cyclic, and the result follows.

Why is angle MCA = DBM? Well, the isosceles trapezium has an axis of (reflective) symmetry, and MCA is the is image of DBM under that reflection. Simple. If we wanted to do it with congruent triangles, this would all be a bit more laborious. First have to show BD=AC using one set of congruent triangles, then CM=BM using another, finally finishing using DM=MA. This is much less interesting. The symmetry of the configuration is a higher-level observation which could be proved from the axioms of geometry if necessary, but gives us more information more quickly. When we use a configuration like the symmedian configuration, we are really doing a higher-again-level version of this.

Anyway, that problem is fine, but it’s not especially difficult.

Consider instead the following problem. (I saw this online, possibly with slightly different notation, a few days ago and can no longer find the link. If anyone can help, I will add the link.)

Let AB be a chord of a circle, with midpoint M, and let the tangents at A and B meet at P. Consider a line through P which meets the circle at C and D in that order. Extend CM to meet the circle again at E. Show DME is isosceles.

Here’s a diagram, though it includes some clues.

I thought this was a fun problem, and for a while I couldn’t do it because despite lots of equal angles and equal lengths, I couldn’t conjure any congruent triangles in the right places, and I didn’t care enough about solving it to get involved in trigonometry. Then came the moment of insight. We have a midpoint, and also the intersection of the tangents. So DP is the symmedian of triangle DAB, and DM is the median. This gives us the two equal orange angles. Cyclicity gives us an extra equal angle at E as well.

Note now that the situation is very very similar to the previous question (after changing some of the labels), only this time we know ACBDE is cyclic, but don’t know that ABDE is an isosceles trapezium. If ABDE is an isosceles trapezium, we are clearly finished, as then by the same symmetry argument, EM=DM. This direction is probably harder to prove than the direction of the previous problem. Again there are a couple of ways to proceed, but one way is to consider the point E’ such that ABDE’ is an isosceles trapezium, and arguing that E’ lies on the given circle, and the circle through BME, and thus must coincide with E, in a reverse reconstruction argument.

Anyway, this is all slightly a matter of taste, but I would say the second problem is much much more fun than the first problem, even though the second part of the solution is basically the first problem but in a more awkward direction. If you’re going to do Euclidean geometry at all (very much another question), I think you should do questions like the second question wherever possible. And the enjoyable ‘aha moment’ came from knowing about the symmedian configuration. Is it really plausible that you’d look at the original diagram (without the dashed orange lines) and think of the antiparallel to AB in triangle DAB through point P? Probably not. So knowing about the configuration gave access to the good bit of a nice problem.

‘Philosophy of this sort of thing’

If the goal was to solve the second problem in a competition, knowing about the symmedian configuration would be a big advantage. I’ve tried to justify a related alternative view that knowing about the configuration gave access to an enjoyable problem. The question is how many configurations to study, and how hard to study them?

We tend not to think of cyclic quadrilaterals as a special configuration, but that is what they are. We derived circle theorems from the definition of a circle so that we don’t always have to mark on the centre, every single time we have a cyclic quadrilateral. So becoming familiar with a few more is not unreasonable. In particular, there are times when proofs are more important than statements. In research (certainly mine), understanding how various proofs work is the most important aspect, for when you try to extend them or specialise. And in lots of competition problems, the interesting bit is normally finding the argument rather than basking in wonder at the statement (though sometimes the latter is true too!).

To digress briefly. In bridge, I don’t know enough non-obvious motifs in bidding or gameplay to play interesting hands well. I trust that if I thought about some of it very very carefully, I could come up with some of them, especially in gameplay, but not in real time. And it is supposed to be fun right?! Concentrating very very hard to achieve a basic level of competence is not so enjoyable, especially if it’s supposed to be a break from regular work. The end result of this is that I don’t play bridge, which is a shame, because I think the hurdles between where I am currently and a state where I enjoy playing bridge are quite low. If I knew I was going to play bridge regularly, a bit of time reading about conventions would be time well spent. And obviously this applies equally in pursuits which aren’t directly intellectual. Occasionally practising specific skills in isolation broadens overall enjoyment in sport, music, and probably everything. As anyone who’s played in an orchestra knows, there are standard patterns that come up all the time. If you practise these occasionally, you get to a stage where you don’t really need to concentrate that hard in the final movement of Beethoven 5, and instead can listen to the horns, make funny faces at the first violins, and save your mental energy for the handful of non-standard tricky bits. And of course, then move on to more demanding repertoire, where maybe the violas actually get a tune.

This is highly subjective, but my view is that in all these examples are broadly similar to configurations in geometry, and in all of them a little goes a long way.

How? In lots of the geometry configurations you might meet in, for example, a short session at a training camp, most of the conclusions about the configurations have proofs which, like in our symmedian case, are simple exercises. Once you’ve got over some low initial experience hurdles, you have to trust that you can normally solve any simple exercise if required. If you can’t, moving on and returning later, or asking for help is a good policy. The proof shown above that symmedians pass through tangent meet points (and especially a trigonometric alternative) really isn’t interesting enough to spend hours trying to find it. The statements themselves are more useful and interesting here. And it can often be summarised quite quickly: “symmedians are the isogonal conjugates of the medians, so they bisect antiparallels, meet at K, and pass through the alternate tangent meeting points.” Probably having a picture in your mind is even simpler.

There’s a separate question of whether this is worthwhile. I think solving geometry problems occasionally is quite fun, so I guess yes I do think it is worthwhile, but I understand others might not. And if you want to win maths competitions, in the current framework you have to solve geometry problems under time pressure. But from an educational point of view, even though the statements themselves have no real modern research value, I think a) that’s quite a high bar to set, and there’s no a priori reason why they should – >99.9% of things anyone encounters before university have no value to modern research maths; b) in terms of knowledge acquisition, it’s similar in spirit to lots of things that are relevant to later study. I don’t have to solve PDEs very often, but when I do, I hope they are equivalent or similar to one of the small collection of PDEs I do know how to solve. If I worked more with PDEs, the size of this collection would grow naturally, after some initial struggles, and might eventually match my collection of techniques for showing scaling limits of random processes, which is something I need to use often, so the collection is much larger. Maybe that similarity isn’t enough justification by itself, but I think it does mean it can’t be written off as educationally valueless.

Balkan MO 2017 Question Two

An acute angled triangle ABC is given, with AB<AC, and \omega is its circumcircle. The tangents t_B,t_C at B,C respectively meet at L. The line through B parallel to AC meets t_C at D. The line through C parallel to AB meets t_B at E. The circumcircle of triangle BCD meets AC internally at T. The circumcircle of triangle BCE meets AB extended at S. Prove that ST, BC and AL are concurrent.

Ok, so why have I already written 1500 words about symmedians as a prelude to this problem? Because AL is a symmedian. This was my first observation. This observation is then a route into non-Euclidean solutions. It means, for example, that you can describe the point of concurrency fairly explicitly with reference to triangle ABC. If you wish, you can then proceed using areal coordinates. One member of the UK team, whom I know is perfectly capable of finding a synthetic solution, did this. And why not? It’s a competition, and if you can see a method that will definitely work, and definitely take 45 minutes (or whatever) then that’s good.

I was taking a break from work in my office, and had no interest in spending the time evaluating determinants because that isn’t enjoyable at any level, so I focused on the geometry.

I think there’s a good moral from the diagram above, which is the first moderately correct one I drew. I often emphasise that drawing an accurate diagram is important, as it increases the chance that you’ll spot key properties. In this case though, where you’re trying to examine a known configuration, I think it’s more important what you choose to include on your diagram, than how accurately you draw it. (In a moment, we’ll see why it definitely wasn’t very accurate.)

In particular, what’s not on the diagram? E is not on the diagram, and S got added later (as did the equal length signs in TB and CS, which rather spoil what’s about to happen). My first diagram was wildly incorrect, but it also suggested to me that the line ST was hard to characterise, and that I should start by deducing as much as possible about S and T by themselves. So by symmetry, probably it was enough just to deduce as much as possible about T.

Label the angles of triangle ABC as <A, <B, And therefore TB is an antiparallel in triangle ABC. (Note this doesn’t look antiparallel on my diagram at all, but as I said, this didn’t really matter.) Obviously you then guess that CS is also an antiparallel, and on a different diagram I checked this, for essentially the same reasons.

We haven’t yet made any use of the symmedian, but this is clearly where it’ll be useful. Note that if we didn’t know about everything in the prelude, we might well have deduced all of this, but we might not have thought to prove that AL bisects TB unless we’d drawn a very accurate diagram.

At this point, we have to trust that we have enough information to delete most of the diagram, leaving just {A,B,C,S,T} and the line AL. There are a few ways to finish, including similar triangles if you try very hard or trigonometry if you do it right, but again knowledge of some standard configurations is useful. Probably the quickest way is to use Ceva’s theorem in triangle ACS. You can also use Menelaus’ theorem in ABC, so long as you know a little bit about where the symmedian meets the opposite side.

An alternative is the following. We have a complete quadrilateral here, namely BTCS, and the intersection of all its diagonals. One is A, one is the proposed point of concurrency, and one is the point at infinity, since TB || CS. You can chase that, but I found it more clear to let P be the intersection of ST and BC (which we want to prove lies on AL), then look at the complete quadrilateral ATPB. Then AT and BP meet at C, and AB and TP meet at S. So if we look at where the diagonals of ATPB meet the line CS, we have a harmonic range.

If I’d wanted, I could instead have written the prelude about harmonic ranges, but I had fewer ideas how to set these up in a slick Euclidean way. Also, it feels better to talk about the start, rather than the end of a proof, especially when there were alternative endings. Anyway, a harmonic range is a collection of two pairs of points on a line (A, B; C, D), satisfying the following ratio of directed lengths:

\frac{AC}{BC} = -\frac{AD}{BD}.

A classic example is when D is the point at infinity, the RHS is -1, and so C is the midpoint of AB. Being happy about using the point at infinity is a property of projective geometry, of which this is a good first example. Anyway, returning to the problem, we are looking at where the diagonals of ATPB meet line CS, and this pair of points forms a harmonic range with (C,S). TB meets CS at the point at infinity, and so AP meets CS at the midpoint of CS. But from the symmedian configuration, AL bisects CS, so AP and AL are in fact the same line, and so P lies on AL as required.

I think was a brilliant example of when knowing a bit of theory is enjoyable. It wasn’t at all obvious initially how to use the symmedian property, but then the observation that TB is antiparallel felt like a satisfying breakthrough, but didn’t immediately kill the problem.

Balkan MO 2017 – Qs 1, 3 and 4

The UK is normally invited to participate as a guest team at the Balkan Mathematical Olympiad, an annual competition between eleven countries from South-Eastern Europe. I got to take part in Rhodes almost exactly ten years ago, and this year the competition was held in Ohrid, in Macedonia. There’s one paper, comprising four questions, normally one from each of the agreed olympiad topic areas, with 4.5 hours for students to address them. The contest was sat this morning, and I’m going to say quite a bit about the geometric Q2, and a little bit about Qs 1 and 3 also. In all cases, this discussion will include most of a solution, with some commentary, so don’t read these if you are planning to try the problems yourself.

I’m not saying anything about Q4, because I haven’t solved it. (Edit: I have solved it now, so will postpone Q2 until later today.)

Question One

Find all ordered pairs of positive integers (x,y) such that

x^3+y^3=x^2+42xy+y^2.

The first thought is that if either of x or y is ‘large’, then the LHS is bigger than the RHS, and so equality can’t hold. That is, there are only finitely many solutions. The smallest possible value of y is, naturally, 1, and substituting y=1 is convenient as then y^2=y^3, and it’s straightforward to derive x=7 as a solution.

Regarding the non-existence of large solutions, you can make this precise by factorising the LHS as

(x+y)(x^2-xy+y^2) = x^2+42xy+y^2.

There are 44 terms of degree two on the RHS, and one term of degree in the second bracket on the LHS. With a bit of AM-GM, you can see then that if x+y>44, you get a contradiction, as the LHS will be greater than the RHS. But that’s still a lot of possibilities to check.

It struck me that I could find ways to reduce the burden by reducing modulo various primes. 2, 3 and 7 all divide 42, and furthermore cubes are nice modulo 7 and squares are nice modulo 3, so maybe that would bring the number of possibilities down. But my instinct was that this wasn’t the right way to use the fact that we were solving over positive integers.

The second bracket in the factorisation looks enough like the RHS, that it’s worth exploring. If we move x^2-xy+y^2 from the right to the left, we get

(x+y-1)(x^2-xy+y^2) = 43xy. (1.1)

Now it suddenly does look useful that we are solving over positive integers, because 43 is a prime, so has to appear as a factor somewhere on the LHS. But it’s generally quite restrictive that x^2-xy+y^2 | 43xy. This definitely looks like something that won’t hold often. If x and y are coprime, then certainly x^2-xy+y^2 and y are coprime also. But actually if x and y have a non-trivial common factor d, we can divide both sides by d^2, and it still holds. Let’s write

x=dm,\quad y=dn,\quad\text{where }d=\mathrm{gcd}(x,y).

Then m^2 -mn+n^2 really does divide 43, since it is coprime to both m and n. This is now very restrictive indeed, since it requires that m^2-mn+n^2 be equal to 1 or 43. A square-sandwiching argument gives m^2-mn+n^2=1 iff m=n=1. 43 requires a little bit more work, with (at least as I did it) a few cases to check by hand, but again only has one solution, namely m=7, n=1 and vice versa.

We now need to add the common divisor d back into the mix. In the first case, (1.1) reduces to (2d-1)=43, which gives (x,y)=(22,22). In the second case, after cancelling a couple of factors, (1.1) reduces to (8d-1)=7, from which (x,y)=(7,1),(1,7) emerges, and these must be all the solutions.

The moral here seemed to be that divisibility was a stronger tool than case-reduction. But that was just this question. There are other examples where case-reduction is probably more useful than chasing divisibility.

Question Three

Find all functions f:\mathbb{N}\rightarrow\mathbb{N} such that

n+f(m) \,\big|\, f(n)+nf(m)

for all m,n\in\mathbb{N}.

What would be useful here? There are two variables, and a function. It would be useful if we could reduce the number of variables, or the number of occurences of f. We can reduce the number of variables by taking m=n, to get

n+f(n) \,\big|\, f(n) [1+n]. (3.1)

From this, we might observe that f(n)\equiv 1 is a solution. Of course we could analyse this much more, but this doesn’t look like a 10/10 insight, so I tried other things first.

In general, the statement that a\,|\,b also tells us that a\,|\, b-ka. That is, we can subtract arbitrary multiples of the divisor, and the result is still true. A recurring trope is that the original b is elegant, but an adjusted b-ka is useful. I don’t think we can do the latter, but by subtracting n^2 +nf(m) from the problem statement, we get

n+f(m) \,\big|\, n^2-f(n). (3.2)

There’s now no m on the RHS, but this relation has to hold for all m. One option is that f(n)=n^2 everywhere, then what we’ve deduced always holds since the RHS is zero. But if there’s a value of n for which f(n)\ne n^2, then (3.2) is a very useful statement. From now on, we assume this. Because then as we fix n and vary m, we need n+f(m) to remain a divisor of the RHS, which is fixed, and so has finitely many divisors. So f(m) takes only finitely many values, and in particular is bounded.

This ties to the observation that f\equiv 1 is a solution, which we made around (3.1), so let’s revisit that: (Note, there might be more elegant ways to finish from here, but this is what I did. Also note, n is no longer fixed as in previous paragraph.)

n+f(n) \,\big|\, f(n) [1+n]. (3.1)

Just to avoid confusion between the function itself, and one of the finite collection of values it might take, let’s say b is a value taken by f. So there are values of n for which

n+b \,\big|\, b(1+n).

By thinking about linear equations, you might be able to convince yourself that there are only finitely many solutions (in n) to this relation. There are certainly only finitely many solutions where LHS=RHS (well, at most one solution), and only finitely many where 2xLHS=RHS etc etc. But why do something complicated, when we can actually repeat the trick from the beginning, and subtract b(n+b), to obtain

n+b \,\big|\, b^2-b.

For similar reasons to before, this is a great deduction, because it means if b\ne 1, then the RHS is positive, which means only finitely many n can satisfy this relation. Remember we’re trying to show that no n can satisfy this relation if b\ne 1, so this is definitely massive progress!

If any of what’s already happened looked like magic, I hope we can buy into the idea that subtracting multiples of the divisor from the RHS is the only tool we used, and that making the RHS fixed gives a lot of information about the LHS as the free variable varies. The final step is not magic either. We know that f is eventually 1. If you prefer “for large enough n, f(n)=1,” since all other values appear only finitely often. I could write this with quantifiers, but I don’t want to, because that makes it seem more complicated than it is. We genuinely don’t care when the last non-1 value appears.

Anyway, since we’ve deduced this, we absolutely have to substitute this into something we already have. Why not the original problem statement? Fix m, then for all large enough n

n+f(m) \,\big|\, 1+nf(m). (3.3)

To emphasise, (3.3) has to hold for all large enough n. Is it possible that f(m)=2? Again, it’s easy to convince yourself not. But, yet again, why not use the approach we’ve used so profitably before to clear the RHS? In fact, we already did this, and called it (3.2), and we can make that work [3.4], but in this setting, because f(m) is fixed and we’re working with variable large n, it’s better to eliminate n, to get

n+f(m)\,\big|\, f(m)^2-1,

again for all large enough n. By the same size argument as before, this is totally impossible unless f(m)=1. Which means that in fact f(m)=1 for all m. Remember ages ago we assumed that f(n) was not n^2 everywhere, so this gives our two solutions: f(n)=1,\, f(n)=n^2.

Moral: choosing carefully which expression to work with can make life much more interesting later. Eliminating as many variables or difficult things from one side is a good choice. Playing with small values can help you understand the problem, but here you need to think about soft properties of the expression, in particular what happens when you take one variable large while holding another fixed.

[3.4] – if you do use the original approach, you get n^2-1 on the RHS. There’s then the temptation to kill the divisibility by taking n to be the integer in the middle of a large twin prime pair. Unfortunately, the existence of such an n is still just a conjecture

Question Four

(Statement copied from Art of Problem Solving. I’m unsure whether this is the exact wording given to the students in the contest.)

On a circular table sit n>2 students. First, each student has just one candy. At each step, each student chooses one of the following actions:

(A) Gives a candy to the student sitting on his left or to the student sitting on his right.

(B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right.

At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps.
Find the number of legitimate distributions.

My moral for this question is this: I’m glad I thought about this on the bus first. What I found hardest here was getting the right answer. My initial thoughts:

  • Do I know how to calculate the total number of possibilities, irrespective of the algorithm? Fortunately yes I do. Marbles-in-urns = barriers between marbles on a line (maybe add one extra marble per urn first). [4.1]
  • What happens if you just use technique a)? Well first you can get into trouble because what happens if you have zero sweets? But fine, let’s temporarily say you can have a negative number of sweets. If n is even, then there’s a clear parity situation developing, as if you colour the children red and blue alternately, at every stage you have n/2 sweets moving from red children to blue and vice versa, so actually the total number of sweets among the red children is constant through the process.
  • What happens if you just use technique b)? This felt much more promising.
  • Can you get all the sweets to one child? I considered looking at the child directly opposite (or almost-directly opposite) and ‘sweeping’ all the sweets away from them. It felt like this would work, except if for some parity reason we couldn’t prevent the final child having one (or more, but probably exactly one) sweets at the crucial moment when all the other sweets got passed to him.

Then I got home, and with some paper, I felt I could do all possibilities with n=5, and all but a few when n=6. My conjecture was that all are possible with n odd, and all are possible with n even, except those when none of the red kids or none of the kids get a sweet. I tried n=8, and there were a few more that I couldn’t construct, but this felt like my failure to be a computer rather than a big problem. Again there’s a trade-off between confirming your answer, and trying to prove it.

Claim: If n is even, you can’t achieve the configurations where either the red children or the blue children have no sweets.

Proof: Suppose you can. That means there’s a first time that all the sweets were on one colour. Call this time T. Without loss of generality, all the sweets are on red at T. Where could the sweets have been at time T-1? I claim they must all have been on blue, which contradicts minimality. Why? Because if at least one red child had at least one sweet, they must have passed at least one sweet to a blue neighbour.

Now it remains to give a construction for all other cases. In the end, my proof has two stages:

Step One: Given a configuration, in two steps, you can move a candy two places to the right, leaving everything else unchanged.

This is enough to settle the n odd case. For the even case, we need an extra step, which really corresponds to an initial phase of the construction.

Step Two: We can make some version of the ‘sweeping’ move precise, to end up in some configuration where the red number of children have any number of sweets except 0 or n.

Step one is not so hard. Realising that step one would be a useful tool to have was probably the one moment where I shifted from feeling like I hadn’t got into the problem to feeling that I’d mostly finished it. As ever in constructions, working out how to do a small local adjustment, which you plan to do lots of times to get a global effect, is great. (Think of how you solve a Rubik’s cube for example.)

Step two is notationally fiddly, and I would think very carefully before writing it up. In the end I didn’t use the sweeping move. Instead, with the observation that you can take an adjacent pair and continually swap their sweets it’s possible to set up an induction.

Actual morals: Observing the possibility to make a small change in a couple of moves (Step one above) was crucial. My original moral does still hold slightly. Writing lots of things down didn’t make life easier, and in the end the ideas on the bus were pretty much everything I needed.

[4.1] – one session to a group of 15 year olds is enough to teach you that the canon is always ‘marbles in urns’ never ‘balls’ nor ‘bags’, let alone both.

EGMO 2017 – Paper One – Geometric subconfigurations

I’ve recently been in Cambridge, running the UK’s annual training and selection camp for the International Mathematical Olympiad. My memories of living and studying in Cambridge are very pleasant, and it’s always nice to be back.

Within olympiad mathematics, the UK has traditionally experienced a weakness at geometry. By contrast to comparable nations, for example those from Eastern Europe, our high school curriculum does not feature much Euclidean geometry, except for the most basic of circle theorems and angle equalities, which normally end up as calculation exercises, rather than anything more substantial. So to arrive at the level required to be in with a chance of solving even the easier such questions at international competitions, our students have to do quite a lot of work for themselves.

I’ve spent a bit of time in the past couple of years thinking about this, and how best to help our students achieve this. The advice “go away and do as many problems as you can, building up to IMO G1, then a bit further” is probably good advice, but we have lots of camps and correspondence training, and I want to offer a bit more.

At a personal level, I’m coming from a pragmatic point of view. I don’t think Euclidean geometry is particularly interesting, even though it occasionally has elegant arguments. My main concern is taming it, and finding strategies for British students (or anyone else) to tame it too [1].

Anyway, I’m going to explain my strategy and thesis as outlined at the camp, then talk about Question 1 from EGMO 2017, a competition held in Zurich this year, the first paper of which was sat earlier today (at time of writing). The UK sent a strong team of four girls, and I’m looking forward to hearing all about their solutions and their adventures, but later. I had intended to talk about the other two questions too, but I can’t think of that much to say, so have put this at the end.

My proposed strategy

Before explaining my proposed strategy, let me discuss a couple of standard approaches that sometimes, but rarely, work at this level:

  • Angle chase (or length chase) forwards directly from the configuration. Consider lots of intersection points of lines. Consider angles and lengths as variables, and try to find relations.
  • Exactly as above, but working back from the conclusion.
  • Doing both, and attempting to meet in the middle.

The reason why this doesn’t work is that by definition competitions are competitive, and all participants could probably do this. For similar reasons competition combinatorics problems tend not to reduce instantly to an exhaustive search.

It’s also not very interesting. I’m certainly unlikely to set a problem if it’s known to yield to such an approach. When students do try this approach, common symptoms and side-effects involve a lot of chasing round conditions that are trivially equivalent to conditions given in the statement. For example, if you’re given a cyclic quadrilateral, and you mark on opposing complementary angles, then chase heavily, you’ll probably waste a lot of time deducing other circle theorems which you already knew.

So actually less is more. You should trust that if you end up proving something equivalent to the required conclusion, you’ll notice. And if you are given a cyclic quadrilateral, you should think about what’s the best way to use it, rather than what are all the ways to use it.

On our selection test, we used a problem which essentially had two stages. In the first stage, you proved that a particular quadrilateral within the configuration was cyclic; and in the second stage, you used this to show the result. Each of these stages by themselves would have been an easy problem, suitable for a junior competition. What made this an international-level problem was that you weren’t told that these were the two stages. This is where a good diagram is useful. You might well guess from an accurate figure that TKAD was cyclic, even if you hadn’t constructed it super-accurately with ruler and compasses.

So my actual strategy is to think about the configuration and the conclusion separately, and try and conjecture intermediate results which might be true. Possibly such an intermediate result might involve an extra point or line. This is a standard way to compose problems. Take a detailed configuration, with some interesting properties within it, then delete as much as possible while keeping the properties. Knowing some standard configurations will be useful for this. Indeed, recognising parts of the original diagram which resemble known configurations (possibly plus or minus a point or line) is a very important first step in many settings.

Cyclic quadrilaterals and isosceles triangles are probably the simplest examples of such configurations. Think about how you often use properties of cyclic quadrilaterals without drawing in either the circle or its centre. The moral is that you don’t need every single thing that’s true about the configuration to be present on the diagram to use it usefully. If you know lots of configurations, you can do this sort of thing in other settings too. Some configurations I can think up off the top of my head include: [2]

  • Parallelograms. Can be defined by corresponding angles, or by equal opposite lengths, or by midpoint properties of the centre. Generally if you have one of these definitions, you should strongly consider applying one of the other definitions!
  • The angle bisector meets the opposite perpendicular bisector on the circumcircle.
  • Simson’s line: the feet of the three perpendiculars from a point to the sides (extended if necessary) of a triangle are collinear precisely when the point is on the circumcircle.
  • The incircle touch point and the excircle touch point are reflections of each other in the corresponding midpoint. Indeed, all the lengths in this diagram can be described easily.
  • The spiral similarity diagram.
  • Pairs of isogonal conjugates, especially altitudes and radii; and medians and symmedians.

Note, all of these can be investigated by straightforward angle/length-chasing. We will see how one configuration turned out to be very useful in EGMO. In particular, the configuration is simple, and its use in the problem is simple, but it’s the idea to focus on the configuration as often as possible that is key. It’s possible but unlikely you’d go for the right approach just by angle-analysis alone.

EGMO 2017 Question 1

Let ABCD be a convex quadilateral with <DAB=<BCD=90, and <ABC > <CDA. Let Q and R be points on segments BC and CD, respectively, such that line QR intersects lines AB and AB at points P and S, respectively. It is given that PQ=RS. Let the midpoint of BD be M, and the midpoint of QR be N. Prove that the points M, N, A and C lie on a circle.

First point: as discussed earlier, we understand cyclic quadrilaterals well, so hopefully it will be obvious once we know enough to show these four points are concyclic. There’s no point guessing at this stage whether we’ll do it by eg opposite angles, or by power of a point, or by explicitly finding the centre.

But let’s engage with the configuration. Here are some straightforward deductions.

  • ABCD is cyclic.
  • M is the centre.

We could at this stage draw in dozens of equal lengths and matching angles, but let’s not do that. We don’t know yet which ones we’ll need, so we again have to trust that we’ll use the right ones when the time comes.

What about N? If we were aiming to prove <AMC = <ANC, this might seem tricky, because we don’t know very much about this second angle. Since R and Q are defined (with one degree of freedom) by the equal length condition, it’s hard to pin down N in terms of C. However, we do know that N is the midpoint opposite C in triangle QCR, which has a right angle at C. Is this useful? Well, maybe it is, but certainly it’s reminiscent of the other side of the diagram. We have four points making up a right-angled triangle, and the midpoint of the hypotenuse here, but also at (A,B,D,M). Indeed, also at (C,B,D,M). And now also at (C,Q,R,N). This must be a useful subconfiguration right?

If you draw this subdiagram separately, you have three equal lengths (from the midpoint to every other point), and thus two pairs of equal angles. This is therefore a very rich subconfiguration. Again, let’s not mark on everything just yet – we trust we’ll work out how best to use it later.

Should we start angle-chasing? I think we shouldn’t. Even though we have now identified lots of potential extra pairs of equal angles, we haven’t yet dealt with the condition PQ=RS at all.

Hopefully as part of our trivial equivalences phase, we said that PQ=RS is trivially equivalent to PR=QS. Perhaps we also wrote down RN=NQ, and so it’s also trivially equivalent to PN=NS. Let’s spell this out: N is the midpoint of PS. Note that this isn’t how N was defined. Maybe this is more useful than the actual definition? (Or maybe it isn’t. This is the whole point of doing the trivial equivalences early.)

Well, we’ve already useful the original definition of N in the subconfiguration (C,Q,R,N), but we can actually also use the subconfiguration (A,P,S,N) too. This is very wordy and makes it sound complicated. I’ve coloured my diagram to try and make this less scary. In summary, the hypotenuse midpoint configuration appears four times, and this one is the least obvious. If you found it, great; if not, I hope this gave quite a lot of motivation. Ultimately, even with all the motivation, you still had to spot it.

Why is this useful? Because a few paragraphs earlier, I said “we don’t know very much about this second angle <ANC”. But actually, thanks to this observation about the subconfiguration, we can decompose <ANC into two angle, namely <ANP+<QNC which are the apex angle in two isosceles triangles. Now we can truly abandon ourselves to angle-chasing, and the conclusion follows after a bit of work.

I’m aware I’ve said it twice in the prelude, and once in this solution, but why not labour my point? The key here was that spotting that a subconfiguration appeared twice led you to spot that it appeared a further two times, one of which wasn’t useful, and one of which was very useful. The subconfiguration itself was not complicated. To emphasise its simplicity, I can even draw it in the snow:

Angle-chasing within the configuration is easy, even with hiking poles instead of a pen, but noticing it could be applied to point N was invaluable.

Other questions

Question 2 – My instinct suggested the answer was three. I find it hard to explain why. I was fairly sure they wouldn’t have asked if it was two. Then I couldn’t see any reason why k would be greater than 3, but still finite. I mean, is it likely that k=14 is possible, but k=13 is not.

In any case, coming up with a construction for k=3 is a nice exercise, and presumably carried a couple of marks in the competition. My argument to show k=2 was not possible, and most arguments I discussed with others were not overwhelmingly difficult, but didn’t really have any key steps or insight, so aren’t very friendly in a blog context, and I’ll probably say nothing more.

Question 3 – Again, I find it hard to say anything very useful, because the first real thing I tried worked, and it’s hard to motivate why. I was confused how the alternating turn-left / turn-right condition might play a role, so I ignored it initially. I was also initially unconvinced that it was possible to return to any edge in any direction (ie it must escape off to infinity down some ray), but I was aware that both of these were too strong a loosening of the problem to be useful, in all likelihood.

Showing that you can go down an edge in one direction but not another feels like you’re looking for some binary invariant, or perhaps a two-colouring of the directed edges. I couldn’t see any way to colour the directed edges, so I tried two-colouring the faces, and there’s only one way to do this. Indeed, on the rare occasions (ahem) I procrastinate, drawing some lines then filling in the regions they form in this form is my preferred doodle. Here’s what it looks like:

and it’s clear that if the path starts with a shaded region on its right, it must always have a shaded region on its right. As I say, this just works, and I find it hard to motivate further.

A side remark is that it turns out that my first loosening is actually valid. The statement remains true with arbitrary changes of direction, rather than alternating changes. The second loosening is not true. There are examples where the trajectory is periodic. I don’t think they’re hugely interesting though, so won’t digress.

Footnotes

[1] – “To you, I am nothing more than a fox like a hundred thousand other foxes. But if you tame me, then we shall need each other. To me, you will be unique in all the world. To you, I shall be unique in all the world,” said the Fox to the Little Prince. My feelings on taming Euclidean geometry are not this strong yet.

[2] – Caveat. I’m not proposing learning a big list of standard configurations. If you do a handful of questions, you’ll meet all the things mentioned in this list several times, and a few other things too. At this point, your geometric intuition for what resembles what is much more useful than exhaustive lists. And if you’re anxious about this from a pedagogical point of view, it doesn’t seem to me to be a terribly different heuristic from lots of non-geometry problems, including in my own research. “What does this new problem remind me of?” is not unique to this area at all!