Math Problems ++

I'm trying to remember the name of a fairly old game I played (from around the time of Starcraft). I know it's not much information to go by, but basically it was a single-player RTS/RPG hybrid game. The defining feature was that you could build units or shops etc. but you couldn't control the units you built. They would walk around and fight on their own. It was set in a fantasy setting. It was 2D with graphics kind of like Starcraft. Any guesses would be appreciated!

EDIT: AHA I figured out it is called Majesty

Please PM me if you are interested in participating in CSL at MIT. I want to gather some data on who is interested.

Include your best iccup rank in your PM. For reference, if you are C-/C level or above we could probably use you for the team. Of course, regardless of your rank CSL can be a great way to meet other Starcraft lovers.

Ok it's my turn to post a shameless homework help thread . In the past I have been really surprised/impressed by the background of folks on this site so I'm hoping you guys can help me out ^^.

I'm trying to write complete solutions to the problems in Ramakrishnan and Valenza's book. The problems are generally approachable but given the sheer volume of questions I know I'm going to get stuck in a few places. The first place I'm getting stuck is on part (b) of problem 6 in Chapter 2.

The book uses the convention that a Banach algebra is a (not necessarily commutative) unital algebra whose elements form a complex Banach space and such that
||a times b|| <= ||a|| times ||b||
for any two elements a and b.

I'm supposed to find an example of a Banach algebra together with two elements a and b such that ab is invertible but ba is not invertible.

It would be awesome if anyone could help!

Disclaimer: These numbers are not supposed to have any real significance

Suppose you are playing an RTS on some ladder. Assume everyone has equal skill levels, so we are simply speaking about build order advantages. Your standard, "safe" build order against a different race gives you an 80% chance of having a 60% chance to win a game. During the other 20% of your games you lose 90% of the time because you are facing a particular cheese.

Your overall chance of winning is 50%. Do you think it likely that you notice and appreciate your 60% natural advantage, or will you assume that you win 60% of the time in long games because of your skill. Do you think it more likely that you feel angry and frustrated because you lose 20% of the time to things outside of your control?

EDIT: Uh I am not saying that skill isn't a factor amongst players. Certainly while laddering you will come across players that are either worse or better than you. However, if you play a random user of equal skill level using a certain build then the above percentages hold. You're mass-laddering using only this one build, facing a bell-curve of skill levels centered around your skill level. The game is balanced, but how do you feel while playing?

While I work on gondolin's interesting problem, here's something along the lines of the problems that have been posted.

Definition: A three-legged spider is a the union of three line segments, all meeting at a single point. For example, a T is a three-legged spider.

Question: Can you fit uncountably many three-legged spiders in the plane? If not, can you prove it is impossible?

EDIT:
The plane is infinite
The spiders need not all be congruent
No leg of any given spider may be contained in another leg of that spider
No two distinct spiders can intersect anywhere

http://www.theitest.com/newspost/view/55

Blergh... I spent 5 consecutive Mondays working through the night to beat people in this thing, and now after the 6th all-nighter I lose it all by some ridiculously slim margin :/

USAMO in a few days!

Harvard waitlisted me... It seems they rejected or waitlisted many of the top US mathematics students of the year (based on math competitions and Westinghouse). 9 of us are going to MIT . I can't help but think Harvard shot itself in the foot.

Here's a problem easier than the last few I've posted:
Given a 1000x1000 chessboard, n squares are colored red and the rest are colored blue. I am not able to pick 3 red squares such that two of them are in different columns and (a possibly different) two of them are in different rows. What is the maximum possible value of n, with proof?

I'm now a solid D+ on iccup but my APM has never been over 90 in a game I've won. I thought it would get better with time but I've been playing for a few months now and it stays put at ~80. I feel I would win a lot more if I were able to move even a little bit faster. Maybe I should just stick to math... the USAMO is only a month away.

Speaking of math, does there exist a countable abelian group with an uncountable chain of nested subgroups? It's a nice problem in the same flavor as some of the roadrunner questions I've seen posted here.

It seems I barely made it into the top 100. I'm happy to make it that far as a high school student, but I can't help feeling I would have placed much higher if I hadn't screwed up A1. Anyone else from teamliquid do well?

EDIT: For Slithe, my favorite problem of the ones I solved:

Let n be a positive integer. Find the number of pairs of polynomials (P(x),Q(x)) with real coefficients such that:
1. P squared plus Q squared is equal to x^(2n) + 1
and
2. The degree of Q is strictly greater than the degree of P

Here, x^(2n) means x to the (2n)th power.

I like it because it has a relatively short, clever solution

EDIT 2: One of my friends who did high school tests with me last year placed in the top 6!

I'm probably just being stupid, but for some reason this seems difficult to me. I'm posting it in a few places on the internet in case someone can help me out. Thanks in advance!

Let M be a connected, compact, smooth manifold without boundary. Let V denote the (real) vector space of all smooth functions from M into the real line. Call a subspace H of V invariant if whenever h is in H and g is a diffeomorphism of M then h composed with g is in H. Prove that the only invariant subspaces are {0}, V, and the set of constant functions on M.