William Lowell Putnam Competition

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William Lowell Putnam Competition

Postby 3.14159265... » Sat Jun 02, 2007 6:11 am UTC

Anyone here write it?
How did you do, any advice for me wanting to write it next year.

I am going through some old ones, they are kinda hard... lol
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Postby skeptical scientist » Sat Jun 02, 2007 6:16 am UTC

I haven't participated in either of the last two years (graduated), but I did for the 3 years before that. I pretty consistently got a score of around 35-40.

My advice is to practice by solving lots of problems (there are sites with long lists of old Putnam problems, and other similar problems, e.g. IMO). Also, it's good to be as broadly educated as possible; while in principle they're generally all solvable with high school math, it never hurts to have some group theory or analysis up your sleeve, or combinatorics, or whatever.
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Postby Cauchy » Sat Jun 02, 2007 10:26 am UTC

In each of the past two years, I answered 7 questions and got 69 of the 70 points, and I think what helped me the most was being able to write succinct, rigorous proofs quickly. Know exactly what constitutes a proof of something, know good ways of proving certain types of statements, know how to write clearly, know effective terminology, know what things you can leave to the grader, and so on. Be familiar enough with this so that when you do solve a problem, you can write up the solution quickly without fumbling around in the details. As you might expect, a good way to practice this is to write up solutions to lots of problems.

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Postby charlesfahringer » Sun Jun 03, 2007 12:04 pm UTC

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Postby Cosmologicon » Sun Jun 03, 2007 6:36 pm UTC

I thought that the test oscillated between easy and hard years, so it surprises me that you three were so consistent. By a huge margin my best score was a 30 (answering, I think, A1, A2, and A3) in a very easy year (meaning the median was 1 rather than 0). I thought that was pretty good, haha. What rank does scoring 69 give you?

I think it would be a huge help to expose yourself to the kinds of tricks that the answers often involve. So be sure to read plenty of solutions as well as questions.

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Postby 3.14159265... » Sun Jun 03, 2007 6:55 pm UTC

I was pretty surprised on the 69 too. (edit: just realized the freudean slip here)

Btw how can I find the scores at my university?
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Postby Buttons » Sun Jun 03, 2007 9:10 pm UTC

I've gotten a 38 for the past two years (woo consistency!), and I agree with everyone else here: read lots of math. For instance, A2 this year was a question about a deterministic game played by two players, Alice and Bob. If you've read a lot about combinatorial game theory, you know exactly how to approach these problems: classify each position as a first player or second player win, and work your way up by noting that a position is a first player win (sometimes denoted an "N-position") if it's possible to move to a second player win position ("P-position"), and a P-position if it's only possible to move to N-positions. After a couple minutes of looking at the small cases, a pattern pops out, and then you've already got the machinery to produce an inductive argument.

I think reading is probably more helpful than practicing (I usually find old exams too disheartening. It's easier to get motivated when you've already invested six hours of your life to doing well.), although it's good to still be in the habit of writing up proofs. If you haven't taken a math course in a while, you might want to write up a few easy problems just to remind yourself how to be rigorous. That said, a buddy of mine who consistently scores in the top 100 carries a practice book with him to glance through in his spare time, and the fact that he regularly kicks my ass is probably related to that fact.

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Postby skeptical scientist » Sun Jun 03, 2007 11:05 pm UTC

3.14159265... wrote:I was pretty surprised on the 69 too. (edit: just realized the freudean slip here)

Btw how can I find the scores at my university?

Whether or not you can find out online depends on your university. You could check the math deparment news page (that's where I found my scores). If they aren't there, you probably have to ask whichever math prof at your school is responsible for proctoring the exam (or if you have a team, ask the coach.)
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Postby Cosmologicon » Mon Jun 04, 2007 12:05 am UTC

Okay, I checked out my university's website. Have you tried a Google search for "putnam scores [university name]"? It was the first hit for me. I was mistaken that my best score was 30; my best score was actually 20:

Code: Select all

year score   rank    %ile
1998   30  410/2581 15.9%
1999   10  729/2900 25.2%
2000   20  245/2818  8.7%


I probably also took it in 2001 but didn't do well enough to be listed. I guess my point is you really can't compare scores from different years, although I'm willing to say that skeptical scientist and Cauchy definitely did much better than I ever did. :)

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Postby 3.14159265... » Mon Jun 04, 2007 12:49 am UTC

Oh, are there other type of competetions for universities too? how do I get invovled
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Postby skeptical scientist » Mon Jun 04, 2007 2:41 am UTC

My scores in Cosmo's format:

Code: Select all

year  score    rank     %ile
2002   37    276/3349   8.2%
2003   40   117.5/3615  3.3%
2004   34   252.5/3733  6.8%

Rank, here, means the average rank of people who had the same score as me, which explains why they are not all integers.

So even though my score was fairly consistent, similar scores had quite different percentile rankings in different years.

While I was browsing through old scores, I noticed that in 2002, some people had scores of 40.9, 40.8, 40.7, etc. I thought scores were always whole numbers of points. Does anyone know what that was about?

By the way, here's an interesting article I found on the exam, for those who don't already know what it is (or those who do, but are interested in what kind of article a reporter for Time magazine would write on the story): Time Magazine article in the archive.
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Postby charlesfahringer » Mon Jun 04, 2007 11:32 am UTC

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Postby Buttons » Mon Jun 04, 2007 7:00 pm UTC

I know someone who got a fractional score once. I think it's part of some tiebreaking procedure to determine the cutoff for the top 100 (when, say, there's a 3-way tie for 99th). I don't know the exact mechanism, but that's why the high-ish scores will sometimes not be integers.

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Postby Cauchy » Mon Jun 04, 2007 11:54 pm UTC

A 69 got me 12th in 2005 and 28th in 2006. I didn't do nearly as well in 2004.

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Putnam Competition?

Postby MiloKam » Wed Nov 28, 2007 7:36 pm UTC

I'm taking part in the Putnam Competition on Saturday and I was wondering if anyone here had any tips or warnings... Anyone?

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Re: William Lowell Putnam Competition

Postby ptveite » Thu Nov 29, 2007 1:33 am UTC

I scored a 31.7 last year, just good enough to break the top 200. I got that taking only the second half of the test (stupid math GRE scheduling), but probably would not have done much better even with the first half, after looking at it. I would say that practicing off of old tests is in some ways useful (get an idea for the problem-solving mentality and general proof techniques) but un-useful in that specific concepts are supposed to never show up on the test twice. As someone else suggested, doing a lot of problem and studying basic proof-writing techniques is probably going to be the most useful.

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Re: William Lowell Putnam Competition

Postby Qoppa » Thu Nov 29, 2007 4:28 am UTC

I'd like to do it after I've got some more math under my belt... That would be cool.

What's the difference between the A and B questions? Also, people keep referring to questions as "the A2 question"; is there some sort of consistency between what A2 (for example) is each year?

Code: Select all

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~/d~/\\b\x7F\177l*~/~djal{x}h!\005h";(++w
<033)?(putchar((*t)(w??(p:>,w?_:0XD)),a()
):0;%>O(x,l)??<_='['/7;{return!(x%(_-11))
?x??'l:x^(1+ ++l);}??>main(){t=&O;w=a();}

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Re:

Postby mudge » Thu Nov 29, 2007 5:36 am UTC

Buttons wrote:A2 this year was a question about a deterministic game played by two players, Alice and Bob.


I was a grader for the guy that wrote that question :)

My advice: Have fun! Most of the questions come down to some trick, so scour your old math books for little weird quirky things and think of how you would use them to write a putnam style question. When you're working on the problems, try coming at the problem in some weird way, or put it into another context, just to get used to manipulating questions around.

Oh, and don't cheat yourself and look at the solutions for old problems unless you've worked on it for like, 3 days at least
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Re: William Lowell Putnam Competition

Postby ptveite » Thu Nov 29, 2007 4:32 pm UTC

There are two sets of questions, 6 problems each, 3 hours each. Section A, you have A1,A2, etc. and likewise with B, there is no substantive difference between A questions and B questions, and there is no real consistency with any given question number from year to year, but in general the lower numbers are easier than the higher ones. In my experience, I can usually do 2, 3, or 4 of A1, A2, B1, B2 and if I'm lucky, one or two of the rest. One piece of advice that somebody else mentioned earlier, and I'd agree with is to do one problem at a time, and make sure you finish that problem. The only scores given out on problems are 0,1,2,8,9,10, and you're only gonna get an 8, 9, or 10 if you have a very nearly complete solution. So submitting a complete, correct for one problem is far preferable to submitting 4 half-assed solutions.

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Re: William Lowell Putnam Competition

Postby Joseph » Thu Nov 29, 2007 5:06 pm UTC

I don't remember my exact scores (I think I pulled a little over 30 for two of the three years I tried it [1997-1999], and a little over 20 for the other), but I know I finished between 200th and 500th each time.

Probably my most humbling Putnam moment was when I tried a B6 problem that read something like this: "Show that for any integers a, b, c we can find a positive integer n such that n3 + a*n2 + b*n + c is not a perfect square. " I had plenty of time after nailing a really easy one and whiffing on the others, so I actually tried to brute force this one, using mod 4 residues.

I don't remember if I got credit for it, but the humbling part came when I read a website that solved it in two lines (link is to problem set itself, not solution).

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Re: Re:

Postby Pathway » Fri Nov 30, 2007 8:24 pm UTC

As are a lot of people, I'm taking it tomorrow--good luck everybody!

Also, that kalva.demon.co.uk site will CRUSH your willpower if you rely on not having a way to look up the solution in order to motivate yourself. It contains nearly all Putnams. Consider yourself warned.

mudge wrote:
Buttons wrote:A2 this year was a question about a deterministic game played by two players, Alice and Bob.


I was a grader for the guy that wrote that question :)


So... you're saying that the guy who wrote this question once took the exam himself, and you graded his Putnam paper way back then?

Wow.
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Re: William Lowell Putnam Competition

Postby 3.14159265... » Fri Nov 30, 2007 10:55 pm UTC

I am taking it tommorow.

Everybody, please do really bad, so I can stand out :D

It should be fun, free lunch, math, coffee perhaps, whats there not to like.

Here is a warmer: If two real numbers x,y are chosen at random from (0,1), using the value Pi, express the probability that the closest integer to x/y is even.

Do not confuse closest integer with roof or floor function.

Oh and ENJOY fellow future mathematicians.
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Re: William Lowell Putnam Competition

Postby Pathway » Sat Dec 01, 2007 12:26 am UTC

3.14159265... wrote:I am taking it tommorow.

Everybody, please do really bad, so I can stand out :D

It should be fun, free lunch, math, coffee perhaps, whats there not to like.

Here is a warmer: If two real numbers x,y are chosen at random from (0,1), using the value Pi, express the probability that the closest integer to x/y is even.

Do not confuse closest integer with roof or floor function.

Oh and ENJOY fellow future mathematicians.


That statement would've been even more awesome if a standard synonym for 'mathematicians' were 'functionologists.'

Also: They give you free lunch and coffee? Where do you go?

Although maybe they do here too. Our adviser hasn't exactly waxed eloquent about test-day procedures.

I'll get back to you on that warmup. 30 seconds isn't enough.
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Re: William Lowell Putnam Competition

Postby 3.14159265... » Sat Dec 01, 2007 12:43 am UTC

Well, I think we are getting lunch (Supervisor's treat).

Coffee was my own, but you know, coffee is worth WAY more than they charge :)
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Re: William Lowell Putnam Competition

Postby Pathway » Sat Dec 01, 2007 12:53 am UTC

Regarding your problem,

Spoiler:
It's fairly easy to see that we want to find the probability that floor[ (x+y)/(2y) ] mod 2 = 0. I don't see how to proceed from here yet.
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Re: William Lowell Putnam Competition

Postby btilly » Sat Dec 01, 2007 1:37 am UTC

3.14159265... wrote:Here is a warmer: If two real numbers x,y are chosen at random from (0,1), using the value Pi, express the probability that the closest integer to x/y is even.


If I didn't do anything silly:
Spoiler:
(5 - Pi)/4

Explanation:
Spoiler:
The nearest integer to x/y will be 0 if x < 0.5y. Which is 1/4. You can see that by integrating P(x < 0.5y | y) = 0.5y over the range y = 0 to 1.

Now what are the odds that x/y > z for some fixed z>1? Well this will be the case if x/z > y. But integrate P(x/z > y | x) = x/z over the range x = 0 to 1 and the probability is 0.5/z.

Now what are the odds that x/y is closest to 2? Well that happens if x/y is between 1.5 and 2.5. The odds that x/y > 1.5 are 1/3, and > 2.5 are 1/5, so the probability is 1/3-1/5.

Now what are the odds that x/y is closest to 4? Well that happens if x/y is between 3.5 and 4.5. The odds that x/y > 1.5 are 1/7, and > 2.5 are 1/9, so the probability is 1/7-1/9.

It is easy to show that the full answer is: 1/4 + 1/3 - 1/5 + 1/7 - 1/9 + 1/11 - 1/13 +...

But that reminds me of a familiar series for Pi/4, which is 1 - 1/3 + 1/5 - 1/7 + ...

So (the answer we want) + Pi/4 = 1/4 + (1/3 - 1/5 + 1/7 - ...) + 1 - (1/3 - 1/5 + 1/7 - ...) = 5/4.

Solve and we get the answer I gave before.

About the main question, I did the Putnam back in the early 90s. I placed in the top 500 each time, but I don't remember where. I think in 1992 I might have been somewhere in the top 100? Someone with access to a math library can look it up in the appropriate edition of The American Mathematical Monthly. (http://links.jstor.org/sici?sici=0002-9890(199210)99%3A8%3C715%3ATFWLPM%3E2.0.CO%3B2-5 suggests Vol. 99, No. 8 (Oct., 1992), pp. 715-724 might be a good place to look.)

PS Oh, and good luck to everyone taking it tomorrow!
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Re: Re:

Postby mudge » Sat Dec 01, 2007 2:25 am UTC

Pathway wrote:As are a lot of people, I'm taking it tomorrow--good luck everybody!

Also, that kalva.demon.co.uk site will CRUSH your willpower if you rely on not having a way to look up the solution in order to motivate yourself. It contains nearly all Putnams. Consider yourself warned.

mudge wrote:
Buttons wrote:A2 this year was a question about a deterministic game played by two players, Alice and Bob.


I was a grader for the guy that wrote that question :)


So... you're saying that the guy who wrote this question once took the exam himself, and you graded his Putnam paper way back then?

Wow.


Nono, he's a professor, and I graded homework he assigned to his students.
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Re: William Lowell Putnam Competition

Postby 3.14159265... » Sat Dec 01, 2007 4:12 am UTC

DAMN, I can't sleep :(.

Stupid Putnam contest, why must you be so much fun.

Its 1, and I am still up. DAMN!
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Re: William Lowell Putnam Competition

Postby MiloKam » Sun Dec 02, 2007 12:31 am UTC

So, if we took the test are we allowed to talk about it? I'm a freshmen and was pretty over my head, but I think I got a couple answers. Though I would love to find out what exactly we were supposed to do for the (3k+1) sum problem... but, again, maybe we aren't supposed to talk about it...

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Re: William Lowell Putnam Competition

Postby SimonM » Sun Dec 02, 2007 12:34 am UTC

MiloKam wrote:So, if we took the test are we allowed to talk about it? I'm a freshmen and was pretty over my head, but I think I got a couple answers. Though I would love to find out what exactly we were supposed to do for the (3k+1) sum problem... but, again, maybe we aren't supposed to talk about it...


I don't think you're supposed to talk about it until after Sun 2nd. At least thats what I read on mathlinks
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Re: William Lowell Putnam Competition

Postby antonfire » Sun Dec 02, 2007 12:35 am UTC

I imagine some people couldn't take it today for religious reasons, so it's probably best to wait a bit.
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Re: William Lowell Putnam Competition

Postby Pathway » Sun Dec 02, 2007 12:38 am UTC

So how'd everyone do? I guess we didn't contract not to talk about the questions afterwards, so we're probably ok if anyone wants to speak in depth.

B1 was kinda easy. I think I had almost a valid proof on A6 too. Otherwise I definitely need to learn a bit more about stuff.
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Re: William Lowell Putnam Competition

Postby 3.14159265... » Sun Dec 02, 2007 1:54 am UTC

I got A3, and B1.

Almost got A2, A4, B5 and A5.

I am mad, If I had another 9 hours, I coulda done it.

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Consider if you only had elements of the 1 residue class and the 2 residue class.
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Re: William Lowell Putnam Competition

Postby BaldAdonis » Sun Dec 02, 2007 2:04 am UTC

3.14159265... wrote:I got A3, and B1....

I think I messed up B2. That triangle solution doesn't work unless f(0)=0, but you know that for some x, f(x)=0, so I could've fixed it. This is the worst part of the Putnam: the long walk home when you pick apart your answers and find the mistakes.

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Re: William Lowell Putnam Competition

Postby 3.14159265... » Sun Dec 02, 2007 2:11 am UTC

Hm... maybe they will think it was obvious :D

I am now starting to get mad over the hyperbola one, I just realized I had seen that proof before.......
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Re: William Lowell Putnam Competition

Postby merc » Sun Dec 02, 2007 4:28 am UTC

I don't remember which numbers were which for the most part, so I'm going to just reference problems descriptively :D

Primarily, does anyone know if the solution for the hyperbola one is actually 'Area=4,' or if that's just the "obvious" wrong answer?

If anyone who didn't take the test is interested, the problem was to find the minimun area of a convex set in the plane that intersects both branches of xy=1, and both branches of xy=-1.

As for solved problems, I was able to solve the prime elements problem, the "n=1111..." problem, the f(n) | f(f(n)+1) problem, at least half of the tangent parabolas problem, and most of the "3k+1" probability*. The rest, mostly part B, were absolutely insane.

*A friend pointed out afterwards that I included an extra factor of k!. Hopefully that warrants some credit.

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Re: William Lowell Putnam Competition

Postby MiloKam » Sun Dec 02, 2007 6:10 am UTC

For A3(the repunit one), did anyone get that there were an infinte amount of f's? I figured that if you multiply n by a power of 10 and then add ("said power of ten" - 1)/9 you would get another larger repunit. Because the repunit is an integer you don't have to worry about decimals... right?

Anyone with an answer for A3, please post! I worked so hard on it but my efforts were really in vain. It was the one about systematicly summing the numbers 1 thru 3k-1 and the probability that none of the sums would be divisable by 3.

The B part was really redic, n'est pas? I tried B3 but I ended up just writing the equation to the power of 2007, which I'm very certain is wrong but I'm wondering how far off I am.

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Re: William Lowell Putnam Competition

Postby Pathway » Sun Dec 02, 2007 6:16 am UTC

merc wrote:I don't remember which numbers were which for the most part, so I'm going to just reference problems descriptively :D

Primarily, does anyone know if the solution for the hyperbola one is actually 'Area=4,' or if that's just the "obvious" wrong answer?

If anyone who didn't take the test is interested, the problem was to find the minimun area of a convex set in the plane that intersects both branches of xy=1, and both branches of xy=-1.


Someone else from my school got 4 as well. He might've proved it by parameterizing the curves using sinh and cosh functions.
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Re: William Lowell Putnam Competition

Postby BaldAdonis » Sun Dec 02, 2007 6:31 am UTC

MiloKam wrote:For A3(the repunit one), did anyone get that there were an infinte amount of f's? I figured that if you multiply n by a power of 10 and then add ("said power of ten" - 1)/9 you would get another larger repunit. Because the repunit is an integer you don't have to worry about decimals... right?

Anyone with an answer for A3, please post! I worked so hard on it but my efforts were really in vain. It was the one about systematicly summing the numbers 1 thru 3k-1 and the probability that none of the sums would be divisable by 3.

The B part was really redic, n'est pas? I tried B3 but I ended up just writing the equation to the power of 2007, which I'm very certain is wrong but I'm wondering how far off I am.



Repunits: There aren't an infinite number, you should just have f(x)=x, and any constant function equal to a repunit. So I guess there are an infinite number, but not the good kind of infinite.

3k+1: Consider arrangements of just the numbers congruent to 1 or 2 mod 3. The rest can go anywhere (except the first position), and the "1 or 2" numbers must be arranged so that you don't have an equal number at any point, and you don't have 3 more of one than the other at any point. Then you see that it has to go 1, 1, 2, 1, 2, 1, ... and alternate from then on. Then just permute the numbers into the right spaces (that's where I messed up).

2007: You should be able to work out a recursion of xn+1 = 4xn -6xn-1, and solve that for a general formula.

I found the B part a lot easier, but that might have been the combination lunch/drinks/cigarette/tea that got me relaxed and into writing math. Solved the first two questions in 30 minutes...

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antonfire
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Re: William Lowell Putnam Competition

Postby antonfire » Sun Dec 02, 2007 6:32 am UTC

merc wrote:Primarily, does anyone know if the solution for the hyperbola one is actually 'Area=4,' or if that's just the "obvious" wrong answer?
Yes 4 is the answer. The solution is a bit more complicated. I'm too lazy to write it up (just minimize a function), but here are sketches for some of the other ones:

Spoiler:
The probability one goes more or less like this: there are k+1 1s, k 0s and k 2s (modulo 3). It doesn't matter where you put the zeros (not quite, see below), so all you care about is the 1s and 2s. You can't have two 1s or two 2s in a row, except at the start. since there are more 1s, than twos, the sequence (without the zeros) has to go 1121212121... . There are (3k choose k) ways to place the zeros (you can't have one at the start), and k!, k!, and (k+1)! ways to order the 0s, the 2s and the 1s, respectively. This gives (3k choose k)*k!*k!*(k+1)! arrangements, out of (3k+1)!, which is a probability of k!*(k+1)!/((2k)!*(3k+1)).

I actually screwed this up by forgetting that you can't put a zero at the start, so I answered k!*(k+1)!/(2k+1)!. I hate making stupid mistakes like that.


For the repunit one, you can use a limit thing to show that it has to take a repunit of length k to a repunit of length ak+b. That determines the polynomial. So, yes, there is an infinite number, but there are even more than just what you mentioned.


The xn+1=3xn+[sqrt(5)*xn] one I almost solved, but I ran out of time. You can find explicit formulas of the form a*(3+sqrt(5))n+b*(3-sqrt(5))n (or something of that shape.. I haven't worked it out) for both xn and [sqrt(5)*xn], prove that it all works out, and just plug n=2007 in.


BaldAdonis wrote:Repunits: There aren't an infinite number, you should just have f(x)=x, and any constant function equal to a repunit. So I guess there are an infinite number, but not the good kind of infinite.
Um. 10x+1?


Anyone know a nice proof for the order p elements of a group one?
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?


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