I am very happy – I scored well enough on the AMC12 and AIME to make the USAMO.

Of course, practically, this isn’t a very good thing. In essence, it means that I get to spend nine hours after school taking a mathematics competition when I could be home doing more productive things. And, while it is certainly impressive to have qualified for the USAMO, I somehow doubt it will be all that useful to put that on my résumé.

But it is still very cool to be able to qualify for such a contest, and perhaps – if I am either extremely lucky or much better at math than I thought I was – I will even be in the top six in the nation and make the American IMO team. I would get to go to Vietnam for a week.

Here’s a sample problem from the 2003 USAMO:

Given a sequence S

_{1}of n+1 non-negative integers, a_{0}, a_{1}, … , a_{n}we derive another sequence S_{2}with terms b_{0}, b_{1}, … , b_{n}, where b_{i}is the number of terms preceding a_{i}in S_{1}which are different from a_{i}(so b_{0}= 0). Similarly, we derive S_{2}from S_{1}and so on. Show that if a_{i}≤ i for each i, then S_{n}= S_{n+1}.

Note that the derived sequence b

_{i}also satisfies b_{i}≤ i (since there are only i terms preceding b_{i}). We show that b_{i}≥ a_{i}for each i. That is obvious if a_{i}= 0. If a_{i}= k > 0, then since each of the first k terms (a_{0}, a_{1}, … , a_{k-1}) must be i ≥ k.Next we show that if b

_{i}= a_{i}, then further iterations do not change term i. If b_{i}= a_{i}= 0, then none of the terms before a_{i}differ from 0, so all the terms before b_{i}are also 0. But that means the corresponding terms of the next iteration must also all be 0. If b_{i}= a_{i}= k > 0, then since a_{0}, a_{1}, … , a_{k-1}all differ from a_{i}, the remaining terms (if any) a_{k}, a_{k+1}, … , a_{i-1}must all be the same as a_{i}. But that implies that each of b_{k}, b_{k+1}, … , b_{i-1}must also be k. Hence if the next iteration is c_{0}, c_{1}, … then c_{i}= k also.Now we use induction on k. Clearly term 0 is always 0. Considering the two cases, we see that term 1 does not change at iteration 1. So suppose that term i does not change at iteration i. If term i+1 does change at iteration i+1, then it must have changed at all previous iterations. So it must have started at 0 and increased by 1 at each iteration.

This will be an… interesting contest. Hopefully I’ll be able to answer at least one of the problems.

Us am O!

We are P!

Congratulations! I did also participate in our national contest but failed. Actually I qualified but since there where too many people and I lost a coin toss I won’t be able to meet you at the IMO. Anyway good luck in the further contest!

Thanks. I took it last week, and haven’t gotten the scores back, but I think I got around a 20/42 – which is fairly good (probably upper half – upper third), but not good enough to qualify for the IMO.

I didn’t want to go to Vietnam anyway. :P