time for Maths

Em(ily), 16, maths nerd. This is where I post my progress and solutions to book problems, bmo problems and other stuff. Most likely 99% frustration.

[alephnowt]

The altitudes from A, B, C of a non-isosceles, acute-angled triangle meet the opposite sides at D, E, F respectively. The line through D parallel to EF meets CA, AB at Q, R respectively, and EF meets BC at P. Prove that the circumcircle of triangle PQR passes through the midpoint of BC.

This is Q7 from this month’s Advanced Mentoring, and the most beautiful geometry I’ve ever seen on the AMS. EVERYTHING IS CIRCLES! This means you can use power of a point over, and over, and over…

[alephnowt]

Here’s the not so nice side of Olympiads. This is working on a Q2 from the AMS (supposed to be easier!) that I just can’t do. I must’ve spent at least 4 hours on it. It’s really frustrating. I’m going to email Gabriel tomorrow for help.

[alephnowt]

A Talk on Dirichlet’s Theorem

Today, I gave my first maths talk ever. I spoke for 20 minutes in front of about 15 people, including a few lecturers and students at Sheff Uni, about some of the cool things I discovered about Dirichlet’s Theorem and its proof.

I was told it was apparently loads better than a lot of the audiences’ first maths talks (!) and that I knew exactly what I was doing and that my voice carried well.

The stuff I was talking about was very nice, although I have no idea how I’d ever come up with anything like that on my own.

I haven’t even mentioned the IMO lecture yesterday!

I’m hoping to get some of the content from the summer project up on this blog in the next few days.

[alephnowt]

I’ve been working on a project on Dirichlet’s theorem recently. This involves trying to prove the theorem from a road map of an elementary proof given in the book ‘A Prime Puzzle’ by Martin Griffiths.

It doesn’t get too difficult to understand, despite some of the content being 3rd-year university material. Such a simple question to state: does every arithmetic progression of integers with coprime step and starting integer contain infinitely many primes?

I’ve found myself just copying out lines I understand when I read them, but not remembering them. I think to finish, I’m going to have to prove the whole thing by myself, using the road map, to show I’ve learned from this project.

The proof is really advanced and really quite elegant, although it’s a little on the fiddly side. To prove the sum we’re interested in tends to infinity, we need to prove it is log x + O(1). The way the book does this involves splitting the sum into two parts, a nice part (log x multiplied by a constant) and a nasty part. The nasty part is chipped away at, piece by piece, and every piece that is chipped off is O(1) until we are left with something that we can also prove is O(1). This chipping and rearranging to knock off a bit more is fiddly and complicated.

I present the project in exactly two weeks’ time. Hopefully, I’ll be ready by then!

[lastoftheicelords]

My mum’s additions to notes on an IMO P6 from a few years ago (additions in orange brackets)

Note the original problem was about landmines, not gold mines :)

[curiosamathematica]

Today I learned about a rather remarkable open problem in mathematics, which looks tantalizingly easy. The question was posed by Ron Graham.

Consider the following recursively defined sequence:

Innocent question: is this sequence unbounded? Surprisingly, the answer to this is unknown—at least according to the source article dating from 2000, Unbounded orbits and binary digits by M. Chamberland and M. Martelli.

(Source: http://www.math.grinnell.edu/~chamberl/papers/mario_digits.pdf)

[szimmetria-airtemmizs]

Here is a very similar problem where we know the answer.

The question is the same, is this sequence bounded or not? In other words, is it possible that the sequence never goes above a certain number?

Try to solve it! It is not easy but you don’t need any kind of advanced tools. 

[alephnowt]

Aww Im glad I made your day. Are you in some kind of university or do you learn all these by yourself?

Nah, just sixth form. There are plenty of training camps to teach us stuff but, in the end, we mostly go it alone. I’m planning on going to university to study maths maths (as opposed to Olympiad maths) but they’re pretty close. Cambridge is my aim :) also pls come off anon if you’re brave enough (I know I pretty much always hide behind the Mafia glasses but it’d be nice to put a blog to the messages!)

[alephnowt]

How can you be so good at maths at the age of just 16? What did you do

Anon I love you, you made my day. :)

In all seriousness, I’ve been damn clever all my life (like I could read at 2/3 years old) and it’s just manifested itself in the maths I do now. For a long time, it kind of lay dormant, but the ability resurfaced when I was about 14/15 and I fell in love with maths. Like I properly, properly absolutely fell head over heels. That’s where the motivation to work my socks off for it comes from. A lot of people have put a lot in to teach me things I know as well, and I’ve flourished from that and from the love that keeps me working for it. :) hope that’s not too pretentious for ya

Maths is awesome --> em loves maths --> em does maths all the time --> em gets good at maths <-- em is pretty smart anyway lol

thats the flow diagram if you like.

[nasa-official]

the moment before your masculinity shatters

[alephnowt]

Um also Venus is the roman goddess of love so :/

[alephnowt]

GUESS WHAT?

My friend Naomi’s decided to apply to Cambridge to do maths!! She was originally going to do Physics which is easier to get into, but then went for Maths with Physics as it’s a better grounding for theoretical physics, then decided in the end that the bits of physics she liked were the maths bits so…. she’s gonna go for maths!! yayyy

[p-locally]

One of these is not like the other…

A problem I’ve been working on in the last few days, it goes like this. Can you reach every point of the plane with integer coordinates starting from  (0, 0) and adding together the vectors (a, b), (b, a), (-a, b), (-b, a)?

It turns out that finding the pairs (a, b) for which you actually can do it is secretly a number theory problem.

I have not entirely solved this problem (I have also asked this question on math.stackexchange wishing for additional input) but I suspect that a sufficient condition would be (a, b) coprime and of different parity.

The gifs above display some of the vectors you can reach starting respectively with the pairs (2, 3), (2, 5) and (1, 3).

(if you want to take a look at my terrible question-writing skills: here’s the link to my math.SE question. I don’t know, it’s really hard for me to concisely and formally describe a mathematical problem so I haven’t been using that site a lot. Also its user-base scares me a little, there are some great mathematicians on there!)

[lastoftheicelords]

I’m assuming from your diagrams that addition kinda includes subtraction, so yes, this seems to be a necessary and sufficient condition! If gcd(a,b) >1, then clearly any point with a coordinate not divisible by this gcd is impossible to reach.

Also if a and b are both odd, if we start at a point whose coordinates are the same parity then adding any of the vectors will give another coordinate with the same parity, so the condition is necessary to get to those with one odd and one even coordinate.

Now let’s show it’s sufficient:

Given the basis shown and field Z for scalars, where can we get to? We can make the vectors (0,2a), (0,2b), (2a, 0), (2b, 0) by adding (b,a) and (-b, a) and similar, and by Bezout’s theorem we can get the vectors (0,2) and (2,0) iff a, b are coprime. From here we only need to get from (0,0) to any coordinate with the parities (o,e), (e,o) and (o,o).

If one of a or b is even and the other odd, wlog let the odd one be a. Then (b,a) and (a,b) are the vectors to get us from (0,0) to (e,o) and (o,e) respectively, and (b, a)+(a,b) gets us to (o,o). From there we can get anywhere using our (0,2) and (2,0).

[alephnowt]

Accidentally posted this to my main blog, but here it is (I'm lastoftheicelords)

[alephnowt]

BaMO 2017 Q3

Find all functions f from the positive integers to the positive integers such that n + f(m) divides f(n) + nf(m) for all n, m positive integers. I will post my solution later, it's such a nice problem. My favourite from this year, and I got it out!

[alephnowt]

BALKANS!!! I won a bronze medal with 21/40 marks. I got two of the four questions out, and picked up a rogue mark on Q4. GBR3 comes home elated :D Our whole team got 4 bronzes, not bad for a UK showing. The team for the Balkans is always a weaker team because the UK rule is that you can only do this competition once. We got an overall score of 98, pretty solid. Sam was one mark away from silver on 30.

[alephnowt]

update from the Balkan MO: I answered Q1 and Q3, so hopefully I'll get a bronze! Fingers crossed :)

[alephnowt]

Balkan Questions

So, I haven't posted the solutions because I haven't written them up, but I have been solving one question a day (....ish) and it's going pretty well! I'd love a medal at the BaMO, this time next week I'll be in the hotel before the flight to Skopje, so excited!!!

[eternalglorytochrist-deactivate]

Math Follow Train

Hey,

I know we math people are often not like the others - and maybe this is a crazy idea - but I think we can do a follow train just like any other community!

If your tumblr is even just partly math-related, reblog this and follow the other people on this train! It is advisable and polite to follow everyone but we’re not gonna be mad if you join the train without following others. I will follow all the people who reblog this :)

You belong on this train if any of the following apply:

* you love math * you study math * you want to study math * you think that math is REALLY COOL * you want to understand math better * you like to talk about math * you have a math-themed blog (even just partly) * you are a mathematician

Yes! I want to do this experiment! Are there enough math-related tumblrs for a successful math follow train!?

[alephnowt]

7/4/17

Does there exist a sequence a_1,a_2,... of positive numbers satisfying both of the following conditions: (i) ∑_{i=1}^{n} a_i ≤ n^2 for every positive integer n; (ii) ∑^{n}_{i=1} 1/a_i ≤ 2008 for every positive integer n?

Sorry about the bad formatting of the question - Tumblr doesn’t like sigma notation or sub/superscript. This was a very nice question, very BMO2-ish. I’m glad I tried a Q2. I read this one when looking around so have had the gist of it in the back of my head for a day or two, so when I sat down to solve it it took 20 minutes at most. I’ve spent less than an hour properly thinking on it in total though, which is a good sign. Mugs of strong tea also really help :)