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:
- GA/GN/CG/GNC/NGA
- AG/CA/NG/GCN/AGA
- 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.))
Note: My 2009 self wrote this. It is preserved for
historical interest and amusement, and as a testament to how far I’d go
to typeset math equations in ASCII art.
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.