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Monday, October 2nd, 2006
1:47 pm - Because sometimes I just want to listen to a nice Raphael.

I think I'm going to need a comforting environment to bob this idea in; I think I'm going to do it here, and then cross-post it to my own journal if I'm comfortable enough.

The reason I thought of this community to post it in, is that I'm going to phrase the idea in terms of a math problem:

What functions, f(x) (reals to reals), have the property that

for all real-valued k,

the sum, with n going from negative infinity to infinity, of f(n+k)

does not depend on k?

In particular, I'm looking for functions f(x) with this property that take a global maximum at 0, are even, are increasing when x is negative and decreasing when x is positive, are always positive-valued, and take a limit of 0 at infinity and negative infinity....

The context:Collapse )

I don't even know who to give this full idea to. Maybe someone knows someone who has a connection....

UPDATE: The Big Picture is now at http://abangaku.livejournal.com/52433.html -- in case anyone, for any crazy and insane reason, would want to take up the jungle-tricky burden to both their artistic and computerial inclinations...!

current mood: excited

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Wednesday, November 23rd, 2005
7:14 pm - the math of textures

Rest assured, a cure for topology has been found and we can now devote ALL our time to inverted category theory!! Our laboratories have tested and retested the horrifying antidote and found it to work on a countably dense category of topological textures, which by the fundamental theorem of topology implies that it works for ALL textures--*especially* the flaky ones. you're all so tall, i kid you not.

[those who live in glass houses should never--under any circumstances--try to kill two birds with one stone]

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Tuesday, March 22nd, 2005
6:46 am - (what is the difficulty level of this problem?)

what is the smallest rectangle that can be formed by placing together an odd number of nonoverlapping P pentominoes? (a P pentomino is the hexagon that it's possible to form by joining a 1*1 square and a 2*2 square along an edge. it's easy to form a rectangle with any *even* number of them....)

current mood: nervous

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Monday, March 21st, 2005
5:29 pm - Putnam Results

Hey. Some of you may have seen this already, but for those who haven't, the Putnam results have come back. MIT repeated as champion, with Princeton runners-up. Duke, Waterloo and Caltech round out the top five, with Harvard in the 6-10 range.
Reid Barton won his fourth Putnam fellow award (big surprise), with Ana Caraiani and Daniel Kane also repeating. Ana also won the Elizabeth Lowell Putnam award for the second straight year. Vladimir Barzov and Aaron Pixton were the other two Putnam fellows. I also saw a few names in the 6-15 set that looked familiar, including Tim Abbott and Alison Miller, but I didn't write them down.

Congrats to all!

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Friday, March 18th, 2005
11:13 pm - two triangles and a point

though i never made MOP (i was close, though, to being a USAMO winner my senior year), i just joined this community today because i'm once again interested in math (after turning away from it for about a year -- my attitude towards it changes periodically), *and* i wanted to post this problem to y'all. oh, and ayo lawrence!

Alfred S. Posamentier gave me this problem (said it was given to him by a guy named Hauptman; i forgot the guy's first name; though the name Hauptman in connection to geometry sounds damn familiar to me):

Given two triangles and a point -- all in the same plane -- prove that there exists a triangle inscribed in one of the triangles that's similar to the other triangle and that has a side (or its extension) passing through the given point.

now, Posmo (as we so affectionately call him) *might* have left out a few details and stipulations, as he stated the problem to me oh so casually and colloquially... it seems like the assertion is false for certain configurations... for example, if the two triangles overlap "a lot" and if the point is located in the intersection of the two sets: 1) the intersection of the interiors of all triangles inscribed in the first triangle and similar to the second; 2) the intersection of the interiors of all triangles inscribed in the second and similar to the first. (it seems easy to have these two sets intersect non-emptily, given sufficiently overlapping triangles to begin with)

so, now, let me ask this, people... aside from this uptight kind of configuration... is the assertion possibly true for "most" configurations? what must we stipulate to have it be true? and hopefully it's an easily stated stipulation... no more than an *eighth* of the way to stipulating the *very* circumstance we desire! (whatever "an eighth of the way" means... though talking in this way is obviously done ironically connecting the euclidean plane of speaking and the hyperbolic one of speaking... *obviously*)

no, and i really *am* serious about this question... i hope someone will have something to say...

current mood: tired

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2:49 am - mathematical addictions

so does anyone perhaps know anything interesting about pentahexes?? (nice friendly wiggly little shapes they are.)

current mood: reborn

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Tuesday, March 15th, 2005
2:13 am - "correct the night's mistakes" -- captain beefheart

so do you think someone could look over this proof i just came up with of a famous graph theory theorem (i don't remember its name)? one, is it accurate and two, if it is accurate, is it a famous proof?

To prove: For n > 0, there are n^(n-2) trees on n distinguishable vertices.Collapse )

current mood: distracted

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Wednesday, February 9th, 2005
11:32 pm - Hi
your_inquisitor Hey, it's Matt Lee of MOP 1997. It's great to find this group; in fact, this is the reason I started a livejournal today. 7.5 years later, I still count MOP among my most meaningful experiences. I wish I could have gone more, but no one invited me to take the AHSME until 11th grade, and maybe I had too much fun and didn't study enough when I was at MOP because I didn't even qualify for the USAMO the following year. I've picked up a bachelor's degree in math from Harvard, will be picking up a master's degree in computer science from UW-Madison, and am trying to figure out what I want to do. abangaku, easwaran, clydej69 - I hope this post finds you well, and I'm looking forward to getting to know the rest of you.

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Monday, November 22nd, 2004
12:58 pm - yet another bizarre semi-mathematical problem

so a straightedge can be used to draw a straight line, a compass can be used to draw a circle, and a piece of string tied around two thumbtacks stuck in a piece of cardboard can be used to draw an ellipse. a piece of thick elastic with a slotted ruler attached end-on to its midpoint can do a pretty good perpendicular bisector, let's say.

what physical device can be used to construct a parabola?

i don't care what you're given... focus and directrix, maybe. preferably, though, the stuff we have is stuff we already know how to construct by physical processes and Euclidean constructions. one thing to make clear -- i don't want the parabola constructed as the locus of anything. i'd like to come up with some process by which an actual continuous curve is drawn on the paper, with hopefully as clear and simple a set of things to consciously worry about (once everything is set up) as for the examples i gave in the first paragraph.

i'd really like to know the answer... heh.

current mood: curious

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Sunday, September 5th, 2004
7:59 pm - All-Time MOP Hall of Fame

The venerable MOP Literature Page (whose URL will soon change) has gotten some updates recently, some of them insignificant. Most notably, Michael Freiman has compiled an all-time MOP hall of fame to showcase the MOP legends who were repeatedly honored in the MOP hall of fame. Bring back memories? Unfortunately, the MOP 2003 Hall of Fame has fallen into the void, and this year's Hall of Fame isn't yet enshrined at Mopper World.

(And somehow the winner of Best Interjection got lost, but it is, indeed, "D'oh". Incidentally, the misdirected ads on the subpages of Zed's MOP '98 page provide good fodder for the easily amused.)

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Wednesday, August 18th, 2004
11:59 am - Bijection Problems

I am interested in finding a source or a book that has a nice section of bijection problems. Problems should range in difficulty from easy AIME problems to easy Olympiad problems. I have the Art of Problem Solving, the Art and Craft of Problem Solving, and Problem Solving Strategies already.

Any help would be appreciated, thanks.

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Sunday, August 1st, 2004
8:57 pm - MOP Pics 2004

The AMC website has MOP pics, even if from the small sample one would conclude that this was GOP, not MOP. (Hmm, I guess that acronym is already taken.) But then, are MOP photos supposed to be a representative sample? And the AMC has again mangled our name... we are now "MOP'ers". Greg looks nice, though. Tedrick put up his photos as well, as linked from the AMC site. (My computer doesn't like the photo album, so I haven't been able to look at them.)

And that endorsement at the bottom of the AMC page is cute. :-) However, Anders has not yet updated his website... perhaps we need a new generation of MOP site-makers.

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Sunday, July 4th, 2004
9:16 pm - Canadian E-Mail Inequality

MOP 2004 has come and gone, and it was incredible as usual. :-)

At the awards ceremony, Jacob Tsimerman, one of the Canadian winners, presented the rest of us with a stronger form of Schur that he'd just received in an email. It runs thus:

a3 + b3 + c3 + 3abc ≥ ab√(2a2 + 2b2) + bc√(2b2 + 2c2) + ca√(2c2 + 2a2)

and for most of MOP, nobody could prove it (although it was verified by TI-89). We kept on writing it on the board before lectures, and on the last night of MOP, Kiran came up a solution, but he needed Maple to take care of the messy algebra.

So here's the challenge: find a reasonably nice solution. Any ideas?

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Saturday, June 19th, 2004
7:26 pm - IMO Team 2004

So after two long days of four and a half hours each of hard, grueling, and annoying problems, a team has emerged. The roster is:

Oleg Golberg
Matt Ince
Tiankai Liu
Alison Miller
Aaron Pixton
Tony Zhang

Alternate: Anders Kaseorg

It'll be a great one!

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Tuesday, March 30th, 2004
9:16 pm - An interesting problem?

I am slightly curious as to how others might solve the following problem:

Let a/b = 2 + sum(k = 1..p-2)((k+1)^(k-1)/k^(k+1)), where a and b are relatively prime positive integers and p is prime. Prove that p divides a.

I created the problem by deriving it from other results, so I have a proof, but it is probably not a very natural proof; I do not know of any very intuitive reason why this result would hold. Any ideas?

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Thursday, March 25th, 2004
12:56 am - APMO 2004

For those who didn't take it, the APMO problems are now online. So we can talk about it here now! What did the people who took it think? Hm... that would actually just be Aaron and Nathan, I think, and I already know how the former did. Well, three's a crowd. The APMO does have a (not entirely deserved) reputation for being an "easy little olympiad", but I think the scores will be particularly high this year. I was once again intimidated by the evil inequality, is all I'll say for now.

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Friday, February 27th, 2004
6:50 pm - Interesting geometry problem I just thought up

Consider the hexagon made by placing a 2*2 square and a 1*1 square flush together (the P-pentomino). This hexagon, and another individual unit square, can be fitted together as two separate pieces in two distinct ways to make the same final figure (an octagon known as the R-hexomino).

What I wonder is, is there any pair of geometrical pieces, planar or no, fractal or no, such that they can be fitted together into the same shape in at least *three* different ways? And if not, can you prove why not? (Different ways are, of course, not counted as different if they are reflections or rotations of each other, though the pieces themselves may be rotated and reflected.)

Any ideas???

current mood: creative

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Sunday, February 1st, 2004
9:14 pm - The Expanding MOP

Some of you already know that MOP will be expanding to 60 people next year: according to this posting the selection process will be the same as the old MOP, only they'll take an extra 30 students in 9th grade and below. I'm sure this will disappoint a lot of 10th/11th graders who were hoping it would be easier to make MOP, but I think it's a pretty good idea. It'll really change the face of MOP, of course (they'll probably insist on stricter supervision and stuff... oh, I miss my wild rookie days when the staff who stayed up late enough to enforce curfew didn't care about it, and vice versa). On the whole I think they'll pull it off better than they did the 2002 MegaMOP -- the only complaint I have is that MOP is only 3 weeks again. Don't they have the money for a longer program? I know it is sort of tough to schedule it, what with the IMO being sort of early, but MOP ran from early June to early July for years...

Okay, enough of my rant. What do other people think?

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1:30 pm - renegade olympiad problem?

okay, so... anyone know a simple proof of the following fact (which i really find fascinating)?

take a cyclic quadrilateral. there are two ways to divide it into two triangles, using a diagonal. prove that the sum of the two inradii of the triangles is the same, no matter what your choice of diagonal is. (you can easily generalize this to any cyclic polygon divided into triangles.)

it's such a simple fact, i can't believe i haven't seen it in any exposition of theorems. but i've only seen complex proofs of it. i hope it generalizes into some big theory, like inversion. that would be exciting.

i also wonder if the converse is true... if the sums of the two inradii are equal, is the quadrilateral cyclic necessarily?

current mood: lazy

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Saturday, January 24th, 2004
5:34 am - introduction

Hey... just thought i'd introduce myself here to all the people out there who don't know me. my name is Lawrence Detlor, and i was at MOP for three years, '97 through '99 (my freshman through junior years of highschool). i'm leaving this morning (!!!) to start what will hopefully be my third-from-last semester at college, at Brown University (it's making me kind of nervous right now!), and otherwise i live in the same Manhattan apartment (so far, at least) as i've lived in since i was 2. let's see... i don't think i'm as much tapped in to olympiad-style math right now as i used to be, since after my 12th grade year started, my interests sort of diverged into various forms of art (as you can see my looking at my journal, especially the "memories" section) -- what would be ideal for me, i think, would be to become a playwright. but i still have all the affection in the world for mathematical concepts, and i love writing poetry in rhyme and meter. i've been thinking about doing a theoretical computer science major at Brown, since i have the hunch that issues of computational complexity underlie all sorts of things in the world. i don't know if i'll have the stomach to go through with it, though, because there are still lots of (what's the opposite of theoretical?) practical comp-sci classes i'd rather not take, if i had the complete choice.

mmm, oh yeah, and i also appointed myself MOP Chronicler of Synchronicity some time back, unearthing such things as the fact that the Coolest Rookie MOP award was won by the fourth-place rookie for (at least) five years in a row, from 1997 through 2001, which is a pattern i just find incredibly stunning and beautiful. i made a guest appearance at MOP 2001 wearing a Harry's Shoes hat, which is how i know landofnowhere, who got me onto this online journaling thing in the first place. anything else? well, enough about me. i'd love to hear about other people on this list, too. keep the MOPping alive for all of us!!

current mood: nervous

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