Putnam 2015 general discussion

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Kent Merryfield

18574 posts

Kent Merryfield

#1 h Dec 6, 2015, 10:10 PM • 5 Y

Y by v_Enhance, champion999, Tawan, Adventure10, Mango247

This topic is for general discussion of the 2015 Putnam exam. How was it for you? Was it harder than usual, easier than usual, or what? What do you think of the mix of problem types? Do you have any predictions about score ranges?

Specific comments about a specific problem should go on the topic for that problem. I'm about to post 12 such topics.

For my own sake, I got reasonable answers to 8 of the problems while proctoring: A1 through A4 and B1 through B4. That doesn't mean I could have completely written up that many answers - maybe something like 5 of them. Compared to my usual reactions, I think that this Putnam was a little on the easier side. Last year, which was a little on the easier side, saw a score of 25 needed to be in the top 500 and 38 to be in the top 200. I think we'll be there again: high 20's for top 500, nearly 40 for top 200.

I personally thought B1 was the easiest question. However, my own students disagree with me on that. A1 was pretty straightforward, not needing much in the way of insight - just careful computation. As was true last year, the difficulty gradient between the easiest and hardest problems seemed fairly steep.

See attachment for all of the problems in one file.

Attachments:

Putnam2015.pdf (74kb)

Putnam2015.tex (3kb)

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v_Enhance

6870 posts

v_Enhance

#2 h Dec 6, 2015, 10:20 PM • 4 Y

Y by mssmath, Tawan, Adventure10, Mango247

I solved eight problems A12345 and B145. Still kicking myself for botching the solution to B3 on the exam; I had all the work I needed but made a careless error at the very end.

I thought that the A test was mostly very easy: problems A1, A3, A4 were all routine, and problem A5 is a little long-winded but nothing magical. Problem A2 was tricky, and took me well over half the A session. I don't know anything about A6, but it seems to be very hard.

But in my opinion the difficulty gradient of the B test was $B4 \ll B1 \ll B5 < B3 < B2$ .

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Wolstenholme

543 posts

Wolstenholme

#3 h Dec 6, 2015, 10:23 PM • 3 Y

Y by Tawan, Adventure10, Mango247

I solved A1234 and B14. Hopefully this will be enough for HM... however I am extremely worried about my solution to problem B1. I made the $g(x) = 8e^{x/2}f(x)$ substitution but then when it sufficed to show that $g'''(x)$ had two distinct real roots I mistakenly cited IVT instead of Rolle's Theorem, having stupidly mixed them up. Do you think my B1 will be graded 0+ or 10-?

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BOGTRO

5818 posts

BOGTRO

#4 h Dec 6, 2015, 10:29 PM • 5 Y

Y by v_Enhance, mssmath, champion999, Tawan, Adventure10

I think the A section was pretty easy and the B section was absurdly hard (or maybe just misnumbered?). A1, A3, A4 were straightforward in that there's really only one thing to try and that thing works without too much effort, though A4 can trick you into thinking it's a nice NT problem instead of just doing the straightforward thing (naive bounding). A2 was trickier in that you either solve it very quickly or probably not at all, but there still weren't a whole lot of false starts there.

On the other hand, B1 seems background-dependent in that if you haven't seen this exact type of problem before, you're gonna have a bad time, and B2 is not entirely obvious what to do (and is also quite difficult to write up). B4 was definitely the easiest of the B set, and B3 wasn't all that bad with sufficient background also.

Anyway this is probably heavily influenced by personal experience; I got A1234 B124 in contest, but a lot of these could apparently be 1s under Putnam grading schemes (compared to e.g. USAMO philosophy where these would all be 7s with maybe 1-2 6s). I guess I have to learn to be more vigilant about proving obvious/well-known stuff; it didn't even occur to me that, for example, the statement "derivative has at least one less root than function" would require a full proof until post-contest discussion.

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AnderExtrema

417 posts

AnderExtrema

#5 h Dec 6, 2015, 10:38 PM • 3 Y

Y by v_Enhance, Tawan, Adventure10

Sadly I slept through part A, I woke up at 9:40 and didn't think I'd be allowed to walk in even if it started at 9. Upon hearing it was the easier part that saddens me a bit.
As for part B, the first one felt simple, sadly I didn't remember linear alg. enough to make it easier. I got B2, although it might no be rigorous enough. The matrix one I'm not sure. After working on it a bit I realized $k$ was allowed to vary to any $k > 1,$ i.e. not just $M^{2}$ at which point I was able to prove a bit about what happens for $k$ odd. I had a lemma which basically stated multiplying a matrix that is in arithmetic progression by an arbitrary matrix is still an arithmetic progression iff the bottom two of the non arithmetic matrice equal.
For the one with triangles, upon seeing the solution I feel dumb. I felt like it had something to do with geomtric series, but I was unsure how to tackle it.
For the permutation one, I tried to get a feel for what was going on with $P_{n}$ but I kept getting lost in counting (I tried using complementary, but still, I just confused myself). Lastly, for B6, all I could figure out was a bit of the growth of A(k) and attempting to look at some pratial sums to see if I could figure out anything.

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pi37

2079 posts

pi37

#6 h Dec 6, 2015, 10:39 PM • 3 Y

Y by v_Enhance, Tawan, Adventure10

I mocked it and solved A1234, B145. Apparently A5 wasn't very hard, but it was very hard for me to get nontrivial time after 4 solutions and 4 writups. In general, though, day A seemed to be on the easier side.

Day B was very weird difficulty. Both 4/5 seemed easier than 2/3, and 1 was only possible for me since I had seen the interlacing trick before (thanks math154). I spent a long time on 2/3 writing out long lists of numbers and trying to write out general forms with diagonalization with no luck.

It seems that each of the 4/5s were below average difficulty especially compared to last year's A5 and B4/B5.

I can't speak for either A6 or B6, but I would bet that a nontrivial number of people fakesolved both: A6 by just dividing by det(M), and B6 by ignoring absolute convergence issues.

Also, a funny note: one of the reasons why I was able to solve B5 very quickly was that I basically wrote it before on an old OMO (posted the link on the specific thread).

This post has been edited 1 time. Last edited by pi37, Dec 6, 2015, 10:40 PM

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djmathman

7936 posts

djmathman

#7 h Dec 6, 2015, 11:02 PM • 8 Y

Y by jh235, Naysh, v_Enhance, mssmath, champion999, Tawan, Adventure10, Mango247

Meh, A123, possibly B1. Why was B so hard

I found A to be pretty easy like others said. A1 was pretty straightforward (you just do it); A2 was a silly problem but used a well-known concept in a way I've never seen before so ok; A3 was pretty straightforward, and the hard part for me was computing the sum of the gcd's.

B was a killer. Spent wayy too much time on B1; never found the $8e^{x/2}$ solution (I used a lemma which stated that if $y$ is a function with at least $n$ distinct roots, $y+y'$ has at least $n-1$ . Then iterating this function three times basically gives the answer, but I'm not sure if my proof of said lemma is up to Putnam standards sooo yea.) I probably should have spent time on B4, as I'm pretty sure I would have gotten it, but I don't think I reasonably could have solved anything else in B.

So basically, if all my solutions are 10s (or maybe one 9), I'm top 200. Otherwise, I'm top 500. Not great, but borderline 200 is pretty much where I consider myself in terms of ability so maybe I shouldn't be too surprised?

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mssmath

977 posts

mssmath

#8 h Dec 6, 2015, 11:56 PM • 3 Y

Y by Tawan, Adventure10, Mango247

I got A123 B134 in like 2.5 hours and had the sketch for A4 that was pretty much vEnhances solution. Man I should have taken this officially, I would have been at least honorable mention. I personally thought B4 was the easiest problem but that was cause I saw the correct solution in like 35 seconds as it is similiar to 2011 AMC #23.

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colinhy

751 posts

colinhy

#9 h Dec 7, 2015, 12:02 AM • 3 Y

Y by mssmath, Tawan, Adventure10

I got A134, B245, albeit on 4 hours of sleep (after an esports session the night before lol) and being late to part B due to a performance. A2 ate up a lot of time (almost 2 hours) and I had the wrong inductive statement for half of that time (whoops). A13 were sightsolve and A4 was straightforward I think. I looked at A5 in the last 5 minutes and got some partial (1 point probably)... kinda wish I spent more time on it instead of A2. B1 was dumb (lol no idea how to do), wrong approach on B3 (got right answer though), B4 was straightforward, and B5 was sightsolve.

The 4/5 problems seemed a lot easier than usual, and the 2's seemed quite hard. Difficulty-wise, I'd say something like:
A1 < A3 = B4 < B2 = A4 = B5 < B1 = A5 < A2 = B3 < A6 = B6

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scaryattack

46 posts

scaryattack

#10 h Dec 7, 2015, 3:10 AM • 5 Y

Y by AnderExtrema, mssmath, devenware, Tawan, Adventure10

This was my first Putnam and my second shot at a math competition. Only about 3 of them completely "stumped" me, while I had some ideas and was able to work toward solutions for the others. That being said, I came out of this one with about 0 points. I can try again next year, though! ...and either way it's a lot of fun.

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Pyrious

3 posts

Pyrious

#11 h Dec 7, 2015, 5:47 AM • 2 Y

Y by Tawan, Adventure10

I solved A123 and B12. For some reason I decided to solve them in order so I got stuck at B3 and never looked at B4 or higher, which I guess is unfortunate because most people are saying that B4/B5 were easier than B2/B3. I'm worried though about several "mistakes" that I made in A3, B1 and B2. This was my first time taking the Putnam so I don't know how the grading goes, and whether/how those mistakes will impact my score.

A3: probably a calculation error at the very end (summing up a*gcd(b) for ab=2015).
B1: same as what Wolstenholme did, I called it IVT instead of Rolle's Theorem. I did this on purpose however, because I knew that Rolle's is a trivial corollary of IVT, and I wasn't 100% sure that "Rolle's Theorem" was the correct name.
B2: I found the example 42015, but I wrongly claimed that it was the first sum to end with 2015 in the sequence (12015 is the first). This fact is not at all relevant to the solution so I'm not sure if I'll get a point off or not.

I also turned in partial ideas for A5 and B3. I wouldn't know what constitutes a 1-point-worthy submission but I guess I'll find out.

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Fedor Petrov

520 posts

Fedor Petrov

#12 h Dec 7, 2015, 11:26 AM • 3 Y

Y by Tawan, Adventure10, Mango247

Do not you feel that there were too many problems about recurrence relations and double counting? And no analysis (except B1, this does not count) or geometry.

This post has been edited 1 time. Last edited by Fedor Petrov, Dec 7, 2015, 7:37 PM
Reason: Spelling corrected

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Kent Merryfield

18574 posts

Kent Merryfield

#13 h Dec 7, 2015, 3:45 PM • 3 Y

Y by Tawan, Adventure10, Mango247

Well, A1 was geometry and calculus. Probably Fedor Petrov doesn't think that counts, either. Yes, there was less analysis here then on many previous Putnam exams.

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Kent Merryfield

18574 posts

Kent Merryfield

#14 h Dec 7, 2015, 7:48 PM • 3 Y

Y by Tawan, Adventure10, Mango247

Pyrious wrote:

because I knew that Rolle's is a trivial corollary of IVT,

MVT, not IVT.

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Pyrious

3 posts

Pyrious

#15 h Dec 7, 2015, 10:22 PM • 3 Y

Y by Tawan, Adventure10, Mango247

Kent Merryfield wrote:

Pyrious wrote:

because I knew that Rolle's is a trivial corollary of IVT,

MVT, not IVT.

Mhm you're right; I really scrambled the theorems well.

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codyj

723 posts

codyj

#16 h Dec 8, 2015, 2:20 AM • 3 Y

Y by jh235, Tawan, Adventure10

Am I the only one who thought B was easier than A

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djmathman

7936 posts

djmathman

#17 h Dec 8, 2015, 2:25 AM • 4 Y

Y by codyj, Tawan, Adventure10, Mango247

^^ Not true; a certain E. Wu also thought B was easier than A.

I mean, it's probably possible to think B is easier if you notice the trick to B1 easily (which I guess is possible if you know what you're doing), not get destroyed by the need to do the earliest problems and solve B4, and then get one of B2 or B3 (probably B3 if you're familiar with Cayley-Hamilton). That's a lot of ifs, though....

This post has been edited 1 time. Last edited by djmathman, Dec 8, 2015, 2:26 AM

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iglenini

50 posts

iglenini

#19 h Dec 8, 2015, 4:46 AM • 3 Y

Y by Tawan, Adventure10, Mango247

I asked this elsewhere but didn't get an answer, so I'll ask this here too. I'm not really familiar with Putnam grading as I'm used to the USAMO, so sorry if this is a really common question:

If for B4, I think I used completely correct logic, but a silly algebra mistake and dropped a factor of 7 to get (17/73)*(15+(4/3))=(833/219) instead of (17/73)*((15/7)+(4/3))=(17/21), do you guys think that this would bring the proof down to a 1-2 level, or would it still be like an 8?

Otherwise, to my knowledge, I solved A12345 and B125, so if I wrote my solutions well enough, I could end up super well.

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KarlMahlburg

23 posts

KarlMahlburg

#20 h Dec 9, 2015, 2:53 AM • 5 Y

Y by jh235, CGGR, Tawan, Adventure10, Mango247

I'm surprised by others' reactions thus far, as I didn't find this to be an overwhelmingly "easy" year. While proctoring, I solved A1, A2, A3, A4, A5, B1, B2, B3, and (almost) B5. Glanced at A6, but was short on time. Every problem except B1 took me at least 20 minutes. Spent some time on B4, but didn't see the correct symmetries. Still haven't looked at B6. Overall, I thought the problems were a strange but enjoyable mix (lots of Number Theory?!), with one glaring exception of a problem that I thought was quite inappropriate. More detailed comments below.

First I'll comment that it is a bit meaningless to talk about the "difficulty" of the Putnam as a single characteristic, as it is a very different experience for students at different levels. I find it more useful to group points/problems into rough categories:
1. "Gimme/Guessable" These are problems that have easy special cases and/or the possibility of educated guesses. This is what dictates the median being 0, 1, or 2.
2. "Accessible" These are problems that most students will at least recognize as familiar and can attempt to approach. These dictate the cutoff for Top 500.
3. "Doable" The problems that at least some (good) students will attempt and solve. These dictate HM; Fellows are too small a small sample size to really guess at.

This year I would group the problems as follows:
Gimme: None - I'm expecting a median of 0, though 1 is possible.
Accessible: A1 and B2. I may revise this, but I expect Top 500 to be about 20-21 points.
Doable: A2, A3, A4, A5, A6 (maybe), B1, B3, B4, B5. Scores for HM and Fellows could be on the high side, say 50 for HM, although my total of 8-9 solutions is typical. Time does become an issue, even when so many problems are doable.

A1 - Fairly involved calculation to solve the optimization problem directly; a bit easier with the geometric insight of the tangent line (which I missed initially). Probably very little partial credit available on this problem.
A2 - Cute example of a linear recurrence that is reversible and symmetric.
A3 - Completely inappropriate Putnam problem! It is fairly trivial and uninteresting if one knows cyclotomic polynomials/fields, but this is a graduate-level topic, and it is hopeless otherwise. The only nice feature of this problem is the multiplicative function.
A4 - Neat problem. Took me 4 tries to find the true boundary case (I looked at x = 0, 1, 1/2, and finally 2/3).
A5 - There has been a minor trend in recent years of easy number theory problems hidden in A4/A5/A6.
A6 - Noticed the invertible case right away, but didn't put together the general solution.
B1 - A little tricky (for B1) in that it effectively requires two substitutions, but this was fast. Again very little partial credit available.
B2 - Fun problem - one doesn't realize how sporadic the sequence is without going out to 200 terms! But the solution doesn't require this anyway, as the relevant subsequence is periodic.
B3 - Used matrix diagonalization; didn't think of characteristic polynomials.
B4 - Turns out to be quite easy, but I didn't see it on Saturday.
B5 - Nice problem. Note that the characteristic polynomial from the problem statement is divisible by $x^2 - 1$ ; dividing this out quickly leads to the fundamental combinatorial recurrence.
B6 - Still haven't looked at this one seriously.

(Last year, my initial reaction was that scores would be high (25+ for top 500), but I changed my mind after discussions on this forum. I'll stick to my first prediction this time, since it turned out to be correct last year!)

This post has been edited 1 time. Last edited by KarlMahlburg, Dec 10, 2015, 4:37 AM
Reason: Fixed comment on B5.

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mssmath

977 posts

mssmath

#21 h Dec 9, 2015, 11:51 AM • 3 Y

Y by Tawan, Adventure10, Mango247

KahlMahlburg, for B2 as was shown on the thread, the sequence is not periodic?

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KarlMahlburg

23 posts

KarlMahlburg

#22 h Dec 9, 2015, 8:50 PM • 2 Y

Y by Tawan, Adventure10

mssmath, I'm not sure what you are asking. Look at the thread for B2 - it is already explained there that the sequence is not periodic. However, each $90k + 75$ does occur, which is enough.

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mssmath

977 posts

mssmath

#23 h Dec 9, 2015, 9:32 PM • 2 Y

Y by Tawan, Adventure10

Oh that is what you're you meant, I then agree.

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kedlaya

153 posts

kedlaya

#24 h Dec 9, 2015, 11:17 PM • 5 Y

Y by KarlMahlburg, GreenKeeper, Tawan, Adventure10, Mango247

A bit later than usual, my Putnam archive

http://kskedlaya.org/putnam-archive/

has been updated with problems and solutions for 2015. As usual, for comments, please send me a direct email rather than a PM, since I don't log into AoPS so frequently. Thanks!

Kiran

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Kent Merryfield

18574 posts

Kent Merryfield

#25 h Dec 10, 2015, 2:45 AM • 3 Y

Y by Tawan, Adventure10, Mango247

KarlMahlburg wrote:

A3 - Completely inappropriate Putnam problem! It is fairly trivial and uninteresting if one knows cyclotomic polynomials/fields, but this is a graduate-level topic, and it is hopeless otherwise. The only nice feature of this problem is the multiplicative function.

It it really restricted to being a graduate level topic? Consider the following question, from a high school contest:

Regular decagon $P_1P_2\dots P_{10}$ is drawn in the coordinate plane, with $P_1$ at $(1,0)$ and $P_6$ at $(3,0).$ If $P_n$ is the point $(x_n,y_n),$ compute the numerical value of the product (where $i^2=-1$ )
$(x_1+y_1i)(x_2+y_2i)(x_3+y_3i)\cdots(x_{10}+y_{10}i).$ Is that really all that different? (And as for interest: you did mention the multiplicative function.)

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KarlMahlburg

23 posts

KarlMahlburg

#26 h Dec 10, 2015, 3:31 AM • 3 Y

Y by Tawan, Adventure10, Mango247

Kent,

I think that is an unreasonable high school question as well! Although at least that case only requires remembering (not understanding) the expansion $x^n - 1 = (x-1)(x-\zeta)\cdots (x-\zeta^{n-1})$ for a complex $n$ -th root of unity $\zeta$ .

Of course this is all subjective... but as a further data point, refer to the A3 problem thread, where a few students and proctors also discuss the confusion that $(z^a)^{1/b}$ is not necessarily the same as $z^{a/b}$ unless one is precise about branch cuts. There is no other problem on this year's Putnam that is subject to such a fundamental misunderstanding of the problem statement, which is the reason for my complaint.

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sqing

41403 posts

sqing

#27 h Mar 12, 2016, 3:11 AM • 3 Y

Y by Tawan, Adventure10, Mango247

v_Enhance wrote:

I solved eight problems A12345 and B145. Still kicking myself for botching the solution to B3 on the exam; I had all the work I needed but made a careless error at the very end.

I thought that the A test was mostly very easy: problems A1, A3, A4 were all routine, and problem A5 is a little long-winded but nothing magical. Problem A2 was tricky, and took me well over half the A session. I don't know anything about A6, but it seems to be very hard.

But in my opinion the difficulty gradient of the B test was $B4 \ll B1 \ll B5 < B3 < B2$ .

Putnam 2015 Aftermath

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