Olympiads!
Nothing like a 3 hour paper to make you realise you don't know anything...
This year, like last year, I sat the chemistry and atrophysics olympiads. Those papers are free for anyone to enter (actually, there might be some nominal fee that your school has to pay, I'm not sure... it won't be much I'm sure, the point is you don't need to pass a test) and so I was able to sit them last year as well. Last year I somehow got gold in both, although in both cases there are things which make that a lot less impressive than it sounds - the astrophysics olympiad (BAAO) only started last year and I'm pretty sure the number of people sitting it was very low, and the chemsitry olympiad goes above gold to roentgenium certificates.
This years highlights from those were a really long, difficult question using NMR data to come up with several structures for organic molecules, and carbon dating Richard III's bones in Chemistry. The BAAO had estimates of the scales of asteroid fields (turns out the odds of successfully navigating one are MUCH better than 3,720 to 1) and a lengthy dissection of some of the movie Interstellar's physics. Overall, these papers felt alright if you could get into the swing of them - the questions were long enough that the 3 hours actually felt short (!) and they were interesting and quite open, especially for the BAAO.
The other two subjects that have Olympiads (that I know of) are Maths and Physics. To get into round one of the Maths Olympiad (BMO1) you need to score a high gold on the Senior Maths Challenge. I narrowly passed the BMO threshold (I think it was 130 out of 160) and sat the paper earlier. This paper was much earlier than the science olympiads, and I don't remember exactly when it was, just that it was gruelling. I'm not an excellent mathematician - I like calculus and vectors, but not number theory or geometry - but it was still entertaining to try and solve the puzzles given. The BMO is always short and brutal in style - 6 questions, the totality of which fit on the back of a postcard. You're then given 3 hours to write out rigorous arguments of as many as possible, each question being worth a maximum of 10 marks. I scored 5/60, which is pretty dismal even though just sitting the paper in principle puts me in the top 500 or so.
Highlights from BMO1 include the puzzles "What is the longest run of consecutive terms in the fibonacci sequence that also are consecutive terms in a quadratic sequence? What are the coefficients in said quadratic sequence?" (The fibonnacci sequence is 1,1,2,3,5... a quadratic sequence is U_n=An^2+Bn+C) And the brutally short: "A number is called charming if it equals 2 or can be written as C=3^i5^j where i,j are non-negative integers. Prove that all natural numbers can be written as a sum of distinct charmng numbers".
Just to show that the last question is not totally impossible, let's prove a similar statement: "All natural numbers can be written as a sum of distinct Fibonacci numbers". We'll let P(n) represent the statement:
"All numbers less than or equal to n can be written as a sum of distinct Fibonacci numbers" and we'll try and show that for all n, P(n) is true. The way to show things like this is using induction. We're off to a great start because, 1,2 and 3 are all fibonacci numbers and so we know that P(1), P(2) and P(3) are all true. 4=3+1, so P(4) is true and 5 is a Fibonacci number so P(5) is true, and so on. To show it for all n, let's use this trick:
Assume P(n) is true.
Now, P(n+1) is true if n+1 can be written as a sum of distinct Fibonacci numbers, because by assumption, we can do it for all numbers below n+1.
Note that we can subtract off a large-enough Fibonacci number from n+1 - we can denote this as F(n+1)= the largest Fibonacci numbers less than or equal to n+1
Re-write n+1 = F(n+1) + R [R stands for "remainder"]
If R is less than F(n+1) then it can be written as a sum of distinct Fibonacci numbers by assumption.
Note that R must be less than F(n+1), because if it weren't, then R+F(n+1) > 2F(n+1) and so n+1 > 2F(n+1) which cannot be true as the ratio between Fibonacci numbers must be less than 2
[Side note: to see why this is true, simply use the defining relationship: U_n+2 = U_n+1 + U_n. Now, if The sequence is an increasing sequence, which we can see it is, then U_n+1 > U_n and so when you add them up you don't quite get 2U_n+1 but a little bit less.]
Since R is a sum of distinct Fibonacci numbers, and F(n+1) is a Fibonacci number that occurs nowhere in that sum, n+1=F(n+1)+R is also a sum of distinct Fibonacci numbers.
Since P(1) is true, and we have just shown that P(n)-->P(n+1), P(n) is true for all natural numbers, n.
QED
In fact, the only property we used of the Fibonacci sequence was the "ratio of 2" bit. So we can generalise and say that all natural numbers can be written as a sum of distinct members of ANY sequence whose members are never seperated by more than a factor of 2 when arranged in increasing order. This is a pretty startling fact, and one of the reasons the BMO is fun.
And lastly... the Physics Olympiad. Some time ago about 12 people from my school sat the first part, which is free to enter (again, there might be a small fee, I mean you don't need to qualify). I think 2 of us got golds and I got entry into the next round. I sat the next round last Thursday. I don't think it went very well - I'm not saying it went terribly, I think I answered two out of 5 questions quite well and another one pretty well. If this was the BMO that would probably have been enough. But this paper was sat by only 50 people who qualified, and the style of the paper is much less intense than the BMO - you're not just given a statement to prove and left to it, you're given a variety of questions ranging from simple plug-and-chug questions which aren't worth many marks to longer descriptions and proofs. And I suspect that my performance was not good enough to get into the next stage (a training camp held at Oxford! And then maybe even the International Final) because I know I made silly mistakes and had almost a complete mind-blank on one question. But you never know, so I'll update when I find out.
Everyone should have sat the paper by now, but I'll hold off on posting "highlights" for a bit.
I think Olympiads are a great thing for people to try and do - they let you practise questions which are on the edge of your ability, they introduce you to interesting ideas in your subject, they provide experience of staring at a problem and not knowing what to do and hopefully experience of that eureka moment when you work out how to deal with a problem. In that respect, I think they're one of the next best things to a "Bayesian Conspiracy" .
There are other obvious advantages as well, of course - a lot of people sitting these will be the best in their class, and these help you to know your limitations and learn to lose. On top of that, sitting a 3 hour paper is probably good practise for university exams. It's good practise as well due to the far less structured nature than A level exams. For instance, I know beforehand that my A level C4 paper is going to be worth 75 marks, and will have 6-8 questions. I know roughly how those questions will be distributed and when sitting the paper I know how many marks each one is worth and can tailor my answer to that. Does any of that help me to know more maths? No. Does it help me to get more marks on my maths test? Yes. And that's a calibration issue that I think Olympiads nicely dodge (The BMO scoring is the least vague - 10 marks for completely correct,regardless of method and generally not many marks for anything that's wrong.)
