Wait, are you serious? Under two weeks left, is that what it’s come to?

What happened to my majestic plans to go over every functional equation I failed on, ever? Or to go through a super-intense geometry-immersion period and actually try to develop some of that crazy “intuition” thing? And I have finals coming up too! I just finished a ludicrous deadline-extended geography project that I am absolutely confident is the crappiest paper of my entire school career so far! And despite a semester of (slacking) classes, my Spanish is still only barely at a usable level! Exclamation marks!

Don’t panic… let’s focus on the positive. I am absolutely prepared with my stationery. I bought three spanking new 0.4mm pens that say “Can write for 1000 meters!” because all of my current ones are annoyingly thick and constantly having almost-but-not-quite run out, plus three new mechanical pencils and enough matching 2B lead to last me through college. All the pencils and lead are Pentel. I haven’t done any research, so if something terrible happens in Argentina I know what company to blame, and I am writing it here so I won’t confuse the brands. Also, in view of what happened to SCH’s carry-on baggage last year (luckily there was no geo on Day 1), I got an extra compass I hope won’t be needed.

While we’re listing all the stuff I have gotten ready:

Yes, it’s official now. I’m on the 2012 International Mathematical Olympiad team bound for Argentina, and if I didn’t make a post about this I would be ashamed to call myself a blogger. So, a little moment of smug self-satisfaction should be justified, I hope? And not to mention, last year’s title of youngest Taiwan contestant is not yet passed? Let’s cue the evil laughter!

…or maybe not.

Here is a simple tabulation of our selection problems:

1. GA/GN/CG/GNC/NGA
2. AG/CA/NG/GCN/AGA
3. GA/GN/CA/GCN/AGC

Algebra x9, Combinatorics x7, Geometry x12, Number Theory x8. In other words a distribution in perfect negative correlation with my estimated ability in each subject. At least, that’s how I’ve always estimated them before about a month ago. Ouch, the last stage was the only one of the three where problem distribution for combinatorics actually reached its fair share. (Alternative interpretation: 2011’s distribution was majorly f123ed up with only one real geometry problem, which just means that this year’s battle will probably be difficult for me. (Alternative alternative interpretation: the evil, nasty, wicked, depraved windmill was actually an outrageous negative for me. Gee, I don’t know how to feel. But I should actually do stuff instead of wildly speculating; let’s get back to the topic.))

I have just come back from a week of continuous mathematics, board gaming, and hitting people with sticks to ensure that they get up in time. Yippee! Those problems sure do get around.

Sample problem: This problem rocks!

Brevity has been chosen over accuracy because the whole point is that you should know this stuff already.

Chapter 2: Basic Topology (+ some Ch. 3)

An isolated point of E is in E but not a limit point of it. E is perfect if it is exactly equal to its set of limit points. Equivalently, it is closed and has no isolated points. Ex. 2.44: The Cantor set is perfect.

A compact set is a set for which every open cover has a finite subcover.

Compactness or compact sets have these properties (with made-up names):

(ported from wiki)

Certain notable problems that I don’t want to look through a zillion pages to find.

• Iran TST 1996, notoriously reposted at least 35 times on AoPS (okay, many of these are actually modifications):

  If \(x, y, z > 0\) then

  \[(xy+yz+zx)\left(\frac{1}{(x+y)^2} + \frac{1}{(y+z)^2} + \frac{1}{(z+x)^2}\right) \geq \frac{9}{4}\]

• ISL 1988 #4: if \(1, 2, \ldots, n^2\) are placed in a \(n \times n\) chessboard, some two adjacent numbers differ by at least \(n\)

Limit + Limit

Cheaply, using an Iverson bracket expression:

\[ \begin{aligned} \lim_{a \to \infty} \lim_{b \to \infty} [a > b] &= 0 \\ \lim_{b \to \infty} \lim_{a \to \infty} [a > b] &= 1 \end{aligned} \]

For more continuity, use \(\frac{a}{a + b}\) instead (Rudin Example 7.2).

Uniform convergence on one limit suffices to allow this exchange, almost by definition.

Limit + Continuity

Let

\[ f_\epsilon(x) = \begin{cases} 0 & \text{if } |x| \geq \epsilon \\ 1 - \left|\frac{x}{\epsilon}\right| &\text{if } |x| < \epsilon \end{cases}. \]

This is a matrix.

\[\begin{bmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \end{bmatrix}\]

It has 2 rows and 3 columns, so it is a \(2 \times 3\) matrix.

Matrix addition and multiplication-by-a-scalar is done componentwise. Matrix multiplication is done trickily; it’s associative, distributive, commutative with scalars, linear, anticommutative-under-transposition.

Vectors are like columns of matrices, or matrices with one column. Except row vectors are rows of matrices.

Very simple to explain: if \(P\) is a statement, \([P]\) is 1 if \(P\) is true and 0 if not. So for example

\[\begin{aligned} \lbrack 1 < 2\rbrack &= 1 \\ \lbrack 1 > 2\rbrack &= 0 \end{aligned}\]

It’s like using a boolean as an integer in C or Python.

It’s useful to keep yourself organized when you’re writing summations, especially if you’re summing across terms with a weird condition or if you need to exchange two sums. It’s also useful for writing pathological functions concisely.