AIME

by Gilbert Keith

So, yaah. I took the aime last Tuesday Dag, the thing was so ghetto. I did problems 1,3 and 4 in 15 minutes. That elevated my spirits a lot. I got stuck on problem 4, however for 30 minutes until I looked for what for they wanted the numbers in. See, I just realised that it helps a lot to read the question in its entireity rather than just thinking about how to tackle it first. Then, I started working on other problems, until I realised that my efforts were futile. I tried 14 for a while, which dealt with triangular pyramids and stuff. Basically, I think I got somewhere with my solution; but, when I verified my work with Richard, it seemed radically different. Well, I just guessed 12 on that one.
There were a couple of questions on there that I could have brute forced. #6 for example, would have been easy:

Let \mathcal{S} be the set of real numbers that can be represented as repeating decimals of the form 0.\overline{abc} where a, b, c are distinct digits. Find the sum of the elements of \mathcal{S}.

I just didn’t want to brute force it and was thinking of elegant ways to solveit. #11 was also pretty easy:

A collection of 8 cubes consists of one cube with edge-length k for each integer k,\thinspace 1 \le k \le 8. A tower is to be built using all 8 cubes according to the rules:

\bullet Any cube may be the bottom cube in the tower.
\bullet The cube immediately on top of a cube with edge-length k must have edge-length at most k+2.

Let T be the number of different towers than can be constructed. What is the remainder when T is divided by 1000?

#12, the trig problem was pretty good. I only realised the trick when I was volunteering for NHS ( .
So, Lessons learnt:

  • First 5 are pretty easy (and, counting is pretty easy to learn)
  • 2-3 problems are brute-forcable, and they should be attempted prior to attempting the impossible.
  • Du intense practise the weekend before the aime.
  • Last, but not the least, Amy didn’t du the Aime! :p
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