17 ms·

University of Tokyo Graduate School Entrance Exam – Mathematics (2016) [pdf]

200 points by v3gas 10y ago

ldjb 10y ago
Note that the linked paper is a mathematics exam that applies to the entire faculty (not just CS). In addition to that exam, one must also take a CS-specific exam. It seems that the Summer and Winter exams are rather different: August 2015: http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-s.pdf http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-s.... February 2016: http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-w.pdf http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-w.... Edit: Looking at the exam guide [0], it seems that it's only if you want to sit the exam in the Summer that you must also take the separate maths exam; the Winter CS paper already includes maths questions so there is no separate maths exam. [0] http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/H29csguide_e.pdf http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/H29csguide_e.pd...
- v3gas 10y ago
  Whoops, I didn't do too much research. I actually thought I linked to that CS-specific exam. I think these look rather hard! (I'm even a last year Bachelor student in Math/CS.)
  - ldjb 10y ago
    I'm glad I'm not the only one who thinks this looks difficult. As someone who studied pure undergraduate CS (the course only included a couple of relatively basic maths units) in the UK, I feel I'd have to do a fair amount of self-study before attempting this exam (even the CS-specific exams are quite tough!).
- fmap 10y ago
  I just looked at a few of the problems and these are really well designed. From what I've seen, there's always a clear progression and a way to get partial credit. Plus the problems themselves are interesting and non-obvious. So honestly I think that this is a great exam. The only problem I see is that 5 hours of exam work in a single day is a bit much, even with a one hour break in the middle. It's probably a logistical problem though, with many prospective students being in town just to take this exam...
- mediocrejoker 10y ago
  I would have a lot of trouble with these CS exams because I can't read Japanese
  - ldjb 10y ago
    You don't need to know Japanese. Each question is available in English on the page following the Japanese version.
- Sargos 10y ago
  This makes me glad I'm not in college anymore. I have an M.S. in Computer Science but even then those questions look very hard and only really relevant to PhDs/Researchers. Maybe I'm just spoiled because software development in real life is so much easier than anything you study in college.
demonshalo 10y ago
Why is it that I find these questions always posed in a way that is MEANT TO make me NOT understand them? Granted, math is not my topic. However, I am interested and want to learn. I don't mean to be a dick, but I just simply find it frustrating that the goal of most mathematicians is to pose some question/conjecture in as few sentences as humanly possible. It really grinds my gears how interesting problems are always abstracted behind confusing vertical specific language. Even when reading a Wikipedia article about some cool problem, you get hit in the face with some abstraction that makes most problems seem far more complicated than they actually are. This is probably not a view shared by many here, but if math problems were to be communicated in more natural ways, far more people would be interested in the sciences. Just my 2 cents! Ps. I understand that this is an M.SC. exam. I am talking about math in general, not this pdf!
- yiyus 10y ago
  I do not agree this is the case. Can you give an example? How would you express these questions "in more natural ways"? (alternatively, feel free to point me to some example with other questions)
- sytelus 10y ago
  The issue is that even a little bit of non-technical language can cause loss of precision. A regular person would love if the technical language was watered down and sprinkled with colorful analogies but this is not often helpful to actually attack the problem. The problems in OP link builds on large pyramid of concepts and abstractions all the way starting from numbers. There is no way out than to walk through this hierarchy of concepts one at a time for many years to finally gain understanding of the problem. Sugar coated explanations might make you feel good but don't actually help work on it.
  - demonshalo 10y ago
    I understand that and agree 100%. However, what I was trying to say is that for a non-technical person who is trying to understand a concept rather than solving a problem, accuracy is not the main concern. Sometimes you just need to understand the line of reasoning "roughly speaking". and as I said at the end, I understand that this is a set of M.SC. exam question and my issue is not with this pdf specifically. Take the problem of Integer Factorization, something that I have personally been fascinated by for years. The problem is so ridiculously simple that a 5 year old could understand it. With that said, google for a few wikipedia articles and see the notation and the way it is presented. The non-mathematician does not care about difference of squares or triangular numbers or what.have.you. It seems to be the case that mathematicians are unable to talk about math without requiring the person they are talking to to have extensive knowledge of the topic. That's IMO why Carl Sagan and Feynman were so great. They could take complex problems and reduce them (read communicate) in a good way that most could understand and reason about without needing to traverse the entirety of the knowledge pyramid. Similarly, that is what Numberphile does so well.
x1798DE 10y ago
Interesting that this is entirely in English. Is this the English language version for foreign students, or are Japanese masters students expected to be fluent in English?
- salesguy222 10y ago
  Not sure about Japan but a very large number of computer scientists also speak English. Due to the youngness of the field, the most important material hasn't been translated from its source language, which is English :)
- ldjb 10y ago
  There is also a Japanese language version: http://www.i.u-tokyo.ac.jp/edu/entra/pdf/archive/16math_j.pdf http://www.i.u-tokyo.ac.jp/edu/entra/pdf/archive/16math_j.pd...
  - lovich 10y ago
    That's interesting. I know Japanese has multiple scripts. Problems 1 and 3 appear to be in a much simpler script that looks less dense. Problem 2 is using whatever script that looks closer to Chinese and is full of much more complicated characters. Is there a reason for that? Did Japan import whatever concept question 2 is about from China and developed the math for the other questions independently?
    - sithadmin 10y ago
      As someone with moderate Japanese literacy, I don't really see the distinction between the problems that you're referencing. All problems here are using the same mix of phonetic and logographic characters, which is typical of modern written Japanese. >Did Japan import whatever concept question 2 is about from China and developed the math for the other questions independently? Very unlikely. Japanese concepts expressed by the borrowed Chinese logographic script (known locally as kanji) are frequently quite different than the meaning that would be expressed in Chinese. Additionally, choosing to write a text with relatively dense usage of kanji characters doesn't necessarily imply anything meaningful. For one, the swap between phonetic script and kanji often plays the role that the empty 'space' character plays in western languages. Sometimes the concepts being discussed include words that are homophones/homographs that are difficult to distinguish between in phonetic script, but are easy to distinguish with kanji. For instance, 'hashi' = chopsticks or bridge depending on emphasis. Aside from use of context it's not necessarily clear what you mean if you spell it out, but there are two distinct kanji for either 'chopsticks' or 'bridge' that are usually known, so it's easier to use those to ward off ambiguity. In other cases, choices between use of phonetic scripts or kanji simply comes down to nothing more than convention.
thearn4 10y ago
Some number theory, linear algebra, calculus of variations, combinatorics, etc. From the applied math side of things, it isn't terribly deep. It's actually right about what I would expect from an applied math student who was good at the undergraduate level and ready to study at a competitive university. For general admissions for any discipline though, I'll agre it's probably a bit rough for non applied math grads. At least it doesn't include any analysis, abstract algebra (groups, rings, fields, etc.), or topology. Which would be murder for non-math majors.
- ajarmst 10y ago
  Yep. My Math is undergraduate Electrical Engineering and Graduate Comp Sci. Some of the stuff I'd have to bone up on because I haven't done it in a few decades, but all the problems were familiar and seemed tractable---seemed appropriate for a recent maths undergraduate. (That said, I'd probably bounce hard off of the time limit, and it's been an awful long time since I did any maths where I wasn't allowed to pull a book down off a shelf.)
- n00b101 10y ago
  Honestly, I thought it was an undergrad entrance exam and was feeling sorry for Japanese high school students, before I checked the title again.
  - tonyedgecombe 10y ago
    Yes, I made that mistake.
- bloaf 10y ago
  I missed the "graduate" part of "graduate school" and was temporarily impressed that Japanese high school students would have to learn eigenvalues/vectors. The second question about rotations seemed right out of an AP calculus class, until I saw the differential equations.
  - startupdiscuss 10y ago
    > I missed the "graduate" part of "graduate school" and was temporarily impressed that Japanese high school students would have to learn eigenvalues/vectors. I had the same experience, followed by a "No wonder Sony produced such great products!" moment.
zulrah 10y ago
And I was angry at my uni,that my cs curriculum contained almost no maths. After reading this paper I realized that I am glad I didn't have to study this. Some calculus would have been useful though, as I am in a machine learning field now
- yardie 10y ago
  That's really shocking to hear. When I was in university CS was almost exclusively math. At the time that was the delineation between CS and computer engineering. One was mostly theoretical expressed through equations. The other was just practical application of EE and software development. And the finance college had the MIS program. They were using java.
  - ldjb 10y ago
    In my personal experience studying undergraduate CS, apart from the dedicated maths modules, we did use mathematical notation for lambda expressions, regular expressions, finite automata, Turing machines and the like. We didn't cover calculus or probability though (I think you were expected to learn that in secondary school). I vaguely recall that we touched on matrices and eigenvalues/vectors. But essentially, we were only taught the maths that we needed to know for CS. And of course there were units that didn't use much maths at all, such as systems development and human-computer interaction.
  - canistr 10y ago
    Your post is equally surprising given that for Waterloo, our undergrad ECE curriculums covered all this material. Arguably, we covered significantly more math (albeit in Electrical Engineering, but there was significant overlap with Computer) than what the CS students did.
wayn3 10y ago
This is really really simple btw, for a grad school entrance exam. Anyone who has successfully studied mathematics for a year should be able to score 100% on this exam.
- kutkloon7 10y ago
  I think most American students would score far below 100%.
- stephancoral 10y ago
  Wow you sure are smart why don't you write an email to the faculty of the highly ranked university that wrote this test to let them know how easy it is? Oh wait, you didn't even realize that this is just one of many tests you take on the examination day and you also must orally defend your undergrad thesis to a panel. But go on, continue telling us how smart you are
  - wayn3 10y ago
    im not telling you how smart i am, im telling you how easy the test is. go on, be butthurt.
    - waynisarapist 10y ago
      Still being a cunt I see. I hope you get raped and no one believes you
      - deleted 10y ago
        [deleted]
- jps359 10y ago
  1st question is Linear Algebra 2nd question is Calc 2+Differential Equations 3rd question is probability/stats an analysis class would probably help as well I guess it's pretty feasible to take all of those in a year, especially if mathematics is your area of study. This wasn't the case in engineering in my experience. In any case, this wouldn't be so bad with a 4 year degree. Not really any harder than other PhD candidacy exams I've looked at
ninjin 10y ago
Goodness me, this brings back memories having taken the entrance exam back in 2009. I am glad that they have removed the "We do not guarantee the accuracy of the English translation of the original Japanese version of the exam" clause. It should be noted that this is only one "session" of a series of exams that you take in a single day. If I remember correctly there were two computer science sessions as well, followed by a presentation of your thesis in front of the faculty once your exams had been graded. After this you had to wait for a few weeks for the final result. It should be noted that I don't find these exams to be particularly difficult, the main difficult lies in the fact that the number of potential topics are drawn from the whole University of Tokyo undergraduate curriculum, so if you studied as an undergraduate at a different school or country -- like me -- you are at a significant disadvantage. What I did was to look at ten years or so of exams to get some statistics on what was likely to be on the exam, then locked myself in my room for the whole of January to cover portions that I had not studied before -- like network protocol specifics. In the end, it worked out for me, but I got somewhat lucky in that there were two statistics questions that I could breeze through to cover for my weaker calculus skills given my old school's focus on discrete mathematics.
- microcolonel 10y ago
  Problem 1.1 seems to be within the realm of casual curiosity for somebody who hasn't even entered undergraduate school. It's basically "I know what a matrix is, and how a matrix multiplies with a vector".
  - ninjin 10y ago
    The general structure of the problems is that they start out with easy questions and get progressively more difficult. If you don't get a single "last question" right, the committee will most likely tear you to pieces. At least this was the common understanding back when I was at Todai. Fun random fact, back in 2009 Todai had their own "brand" of toilet paper made from recycled paper -- old archived exams that is. I think that I still have a roll stashed away in a box somewhere. Using it did give you a bit of a weird feeling knowing the anguish old students must have gone through.
ajarmst 10y ago
Hah! I gave 3.1 - 3.4 as an assignment to second-year CompSci students literally three weeks ago. Although my version was to determine the values experimentally for n<=8 bits and then estimate values for n up to 64. Bonus marks awarded for a mathematical proof of the pattern (which isn't really all that bad if you apply probability to patterns of two bits).
- danielweber 10y ago
  What form does the answer to 3.3 look like? Would it break apart the case for abs(r-s) > 1, which is 0?
tasey 10y ago
Wow, people saying that it is "really simple". I feel dumb.
- Insanity 10y ago
  I think that if you have just passed your undergrad degree, these questions are indeed not _that_ hard, because the stuff is still quite fresh in your mind. If you have left university for some time, chances are you did not do any math in that time inbetween, if you are a software engineer. Software engineering requires logical thinking in a different way than mathematics, so we do not really keep our mathematics skills up to date. Looking at document, I can remember touching the necessary mathematics to solve this, but not having done any mathematics of the kind makes me too rusty in mathematics to be able to solve it. With a bit of revision, someone having seen this somewhere in undergrad will probably be able to solve it.
  - ldjb 10y ago
    An undergraduate degree in maths, sure. But if your degree is in computer science, I think it's likely you'd never come across any of this during the course. Some universities may teach it, but there are probably plenty that don't.
    - whorleater 10y ago
      Any self respecting university that teaches a formal computer science degree would teach this level of mathematics. Mine went beyond and happily forced students to learn introductory quantum mechanics as well, which annoyed the CS students to no end.
      - icebraining 10y ago
        I agree; my undergraduate course (not in the US) wasn't even called Computer Science, it was "Software Engineering", but we definitively learned enough math to pass that test easily after the second year.
      - 10y ago
MariuszGalus 10y ago
Question 1. Linear Algebra Level Question 2. Calculus II Level Question 3. Probability & Statistics Level You learn how to solve all of this in Undergraduate American Engineering courses.
- RodericDay 10y ago
  Yep. I was going to say the same thing, seems like a pretty solid test. Series and linear algebra, calculus, probability sounds like just about everything I saw in my Mechanical Engineering degree.
tokyoooo 10y ago
In problem 2, the Euler Lagrange equation imply that we are looking for the function y(x) such as the surface is minimal, so the solution is a cylinder, ie y(x)=constant=c=2.
- piefoot 10y ago
  The solution to the Euler equation gives the minimal surface with is obtained when y(x) is a catenoid, y=c cosh(x/c) here c satisfies 2=c cosh(1/c). The Euler equation gives the minimal distance when the function F is the length of the arc.
ajeet_dhaliwal 10y ago
I believe I could have done all of these questions 12-16 years ago. Shockingly I can't do any of them anymore without refreshing myself for what they are even asking in some cases.
- deleted 10y ago
  [deleted]
kuon 10y ago
I would have been able to do this with no effort 20 years ago, but nowadays I would be totally incapable of solving even one of the problem. Damn I feel the years.
- tluyben2 10y ago
  Same feeling here. I only kept up with discrete math and the rest seems completely lost...
_-__--- 10y ago
I get stuck after the first few subproblems on all three. Does anyone have a PDF of the solutions? Are the solutions ever released? I'm a little rusty on my linear algebra.
- ldjb 10y ago
  Unfortunately, the page the PDF is linked from [0] does not provide solutions. I don't think they are available (officially, at least). [0] http://www.i.u-tokyo.ac.jp/edu/entra/examarchive_e.shtml http://www.i.u-tokyo.ac.jp/edu/entra/examarchive_e.shtml
- Smaug123 10y ago
  I whipped up a quick answer to q2; I don't know if it's right. https://www.patrickstevens.co.uk/misc/TokyoEntrance2016/TokyoEntrance2016.pdf https://www.patrickstevens.co.uk/misc/TokyoEntrance2016/Toky...
rotskoff 10y ago
No algebra at all---seems very odd because much of modern mathematics relies heavily on group and field theory.
- kbart 10y ago
  Maybe they teach advanced algebra during graduate studies, so you are not expected to know it at the entrance exams? At least in my experience that was the case -- I've only learned group and field theory while doing my CS master.
  - ninjin 10y ago
    Todai has a strong traditional science profile and traditionally science deals mainly with calculus and statistics. This leads to less of an emphasis on algebra for these exams since they are based on the undergraduate curriculum. It is regrettable, especially for those like myself that came from a school that emphasised algebra and theoretical computer science and only taught advanced calculus at a graduate level.
- sidusknight 10y ago
  This is applied math. There isn't any analysis, topology, etc, either.
m23khan 10y ago
Assuming it is directed towards admission into a Graduate Mathematics / Engineering Program, it looks pretty average to me. For a student who is in-form (e.g. a fresh graduate or a mature student who has taken 2-3 advanced calculus and algebra+stats courses as refresher), they should be able to perform.
0xfaded 10y ago
I'm a CS grad who is fairly strong with maths but never really did learn to write a proper proof. All of these questions I would have been able to "solve", but the proofs would have been very casual. What would be a valid answer it 2.3 where it basically asks you to integrate an expression of F after substituting y'=dx/dy?
- Smaug123 10y ago
  "2.3" is a bit ambiguous as a label; do you mean part 3 of question 2, or the part of question 2 in which equation 2.3 occurs? If you mean part 2 of question 2: my answer would be "dF/dx = del F/del y dy/dx + del F/del y' dy'/dx; but we know del F/del y = d/dx(del F/del y'), so dF/dx = d/dx(del F/del y') dy/dx + del F/del y' dy'/dx, which by the product rule is d/dx(y' del F/del y'). Now integrate both sides." If you mean part 3 of question 2: just substitute F(y, y') from (2.2) into (2.4).
  - mwane 10y ago
    What is the difference between d and del? > but we know del F/del y = d/dx(del F/del y') ????
    - Smaug123 10y ago
      del refers to the partial derivative, with LaTeX symbol \partial. d refers to the total derivative. "del F/del y = d/dx(del F/del y')" is just the Euler-Lagrange equation.
      - mwane 10y ago
        Which means there is no real difference; if all the types are known, it's just some form of Hungarian notation?
        piefoot 10y ago
        if a function depends only on multiple variables then those are partial derivatives, if the function depends only on one variable then there is only one derivative d/dx
hokkos 10y ago
What exam is it relative to that : https://en.wikipedia.org/wiki/Education_in_Japan#School_grades https://en.wikipedia.org/wiki/Education_in_Japan#School_grad... ? What are the age of the students ?
- ldjb 10y ago
  It's the entrance exam for graduate school, so candidates would likely be 21+.
- TorKlingberg 10y ago
  This is an entrance exam for graduate school, so around age 22–23.
  - hokkos 10y ago
    Wow seems really easy for that age.
    - doall 10y ago
      Yes, the graduate school paper exam of Todai is not difficult compared to the undergraduate entrance exam for that age. The graduate one only requires the very basics to do research. Requirements are different.
dysoco 10y ago
Does anyone have links to Undergraduate entrance exams from other Universities? I am currently taking one in my country and would like to compare the level with mine.
- ldjb 10y ago
  I don't know about other countries, but here in the UK, most universities don't require an entrance exam to study computer science (they go by the grades you leave secondary school / sixth form with). The only two universities I'm aware of that have entrance exams for CompSci are Oxford and Cambridge (because they have to be different): Oxford: http://www.cs.ox.ac.uk/admissions/undergraduate/how_to_apply/MAT_test.html http://www.cs.ox.ac.uk/admissions/undergraduate/how_to_apply... Cambridge: https://www.cl.cam.ac.uk/admissions/undergraduate/admissions-test/ https://www.cl.cam.ac.uk/admissions/undergraduate/admissions...
- piymis 10y ago
  For India, we have entrance exams for admissions into IIT's. Here (http://cms.fiitjee.co/Resources/DownloadCentre/Document_Pdf_42.pdf http://cms.fiitjee.co/Resources/DownloadCentre/Document_Pdf_...) is one solved maths paper of it. Mind you, this is for admission in undergraduate courses straight out of school.
youareanelitist 10y ago
The elitist cunts that are in this chat are the reason why science will never matter to the world. But hey, at least you are better than everyone else right?
Daishiman 10y ago
For a grad school entrance exam this isn't too difficult. It's first and second-year maths stuff for a decent CS degree that doesn't skip out on the basics.
mrcactu5 10y ago
this is graduate school of information science, not the mathematics department of University of Tokyo here's the exam (in Japanese) http://www.ms.u-tokyo.ac.jp/kyoumu/documents/26english.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/documents/26english.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/docs/26a%200119.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/docs/26a%200119.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/docs/26b%200119.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/docs/26b%200119.pdf
nscalf 10y ago
So I never really got too deep into math, once I decided I wasn't going into physics as a career I backed off, but this throws me back into 12th grade AP Physics exams. I'll never forget refusing to take a bad grade on a matrix problem(?) and skipping the next few classes to spend hours trying to do it the long way... I didn't get credit for that question. That being said, now that I'm on the other side of college and in a steady career, I'm really interested in diving deeper into math. Does anyone have any recommendations on resources for learning from Calculus on? I could probably do with a Calc refresher as well. Thanks!
- jimsojim 10y ago
  here's a post from mathstackexchange that I always get back to: http://math.stackexchange.com/questions/843697/learning-higher-mathematics-on-your-own http://math.stackexchange.com/questions/843697/learning-high...
khana 10y ago
Damn. I guess I ain't goin' to grad school.
nmca 10y ago
Interesting to see how it compares to my undergrad efforts: http://s000.tinyupload.com/index.php?file_id=00312832089149851850 http://s000.tinyupload.com/index.php?file_id=003128320891498...
piefoot 10y ago
Solving 3.1 I think the solution is binomial(n1+r-1,r-1) where n1=n-r, but I obtained that formula by calculating the formula for r=1, then r=2, and the relation f(r,n) = sum(f(r-1,n-i),i,0,n) and using maxima with simpsum and factor to obtain a simplified expression whose form suggest the general rule, but I don't see a simpler way.
- Smaug123 10y ago
  If you want something to Google, the result is completely standard, and it goes by the name of "stars and bars".
ejanus 10y ago
I have not truly figured out the difference between permutations and combinations. I know about permutations having 'order' but when I see real questions I got lost. Could someone help this 'lost' soul?

200 points by v3gas 10y ago

ldjb 10y ago

Note that the linked paper is a mathematics exam that applies to the entire faculty (not just CS). In addition to that exam, one must also take a CS-specific exam. It seems that the Summer and Winter exams are rather different: August 2015: http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-s.pdf http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-s.... February 2016: http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-w.pdf http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-w.... Edit: Looking at the exam guide [0], it seems that it's only if you want to sit the exam in the Summer that you must also take the separate maths exam; the Winter CS paper already includes maths questions so there is no separate maths exam. [0] http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/H29csguide_e.pdf http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/H29csguide_e.pd...

v3gas 10y ago
Whoops, I didn't do too much research. I actually thought I linked to that CS-specific exam. I think these look rather hard! (I'm even a last year Bachelor student in Math/CS.)
- ldjb 10y ago
  I'm glad I'm not the only one who thinks this looks difficult. As someone who studied pure undergraduate CS (the course only included a couple of relatively basic maths units) in the UK, I feel I'd have to do a fair amount of self-study before attempting this exam (even the CS-specific exams are quite tough!).
fmap 10y ago
I just looked at a few of the problems and these are really well designed. From what I've seen, there's always a clear progression and a way to get partial credit. Plus the problems themselves are interesting and non-obvious. So honestly I think that this is a great exam. The only problem I see is that 5 hours of exam work in a single day is a bit much, even with a one hour break in the middle. It's probably a logistical problem though, with many prospective students being in town just to take this exam...
mediocrejoker 10y ago
I would have a lot of trouble with these CS exams because I can't read Japanese
- ldjb 10y ago
  You don't need to know Japanese. Each question is available in English on the page following the Japanese version.
Sargos 10y ago
This makes me glad I'm not in college anymore. I have an M.S. in Computer Science but even then those questions look very hard and only really relevant to PhDs/Researchers. Maybe I'm just spoiled because software development in real life is so much easier than anything you study in college.

demonshalo 10y ago

Why is it that I find these questions always posed in a way that is MEANT TO make me NOT understand them? Granted, math is not my topic. However, I am interested and want to learn. I don't mean to be a dick, but I just simply find it frustrating that the goal of most mathematicians is to pose some question/conjecture in as few sentences as humanly possible. It really grinds my gears how interesting problems are always abstracted behind confusing vertical specific language. Even when reading a Wikipedia article about some cool problem, you get hit in the face with some abstraction that makes most problems seem far more complicated than they actually are. This is probably not a view shared by many here, but if math problems were to be communicated in more natural ways, far more people would be interested in the sciences. Just my 2 cents! Ps. I understand that this is an M.SC. exam. I am talking about math in general, not this pdf!

yiyus 10y ago
I do not agree this is the case. Can you give an example? How would you express these questions "in more natural ways"? (alternatively, feel free to point me to some example with other questions)
sytelus 10y ago
The issue is that even a little bit of non-technical language can cause loss of precision. A regular person would love if the technical language was watered down and sprinkled with colorful analogies but this is not often helpful to actually attack the problem. The problems in OP link builds on large pyramid of concepts and abstractions all the way starting from numbers. There is no way out than to walk through this hierarchy of concepts one at a time for many years to finally gain understanding of the problem. Sugar coated explanations might make you feel good but don't actually help work on it.
- demonshalo 10y ago
  I understand that and agree 100%. However, what I was trying to say is that for a non-technical person who is trying to understand a concept rather than solving a problem, accuracy is not the main concern. Sometimes you just need to understand the line of reasoning "roughly speaking". and as I said at the end, I understand that this is a set of M.SC. exam question and my issue is not with this pdf specifically. Take the problem of Integer Factorization, something that I have personally been fascinated by for years. The problem is so ridiculously simple that a 5 year old could understand it. With that said, google for a few wikipedia articles and see the notation and the way it is presented. The non-mathematician does not care about difference of squares or triangular numbers or what.have.you. It seems to be the case that mathematicians are unable to talk about math without requiring the person they are talking to to have extensive knowledge of the topic. That's IMO why Carl Sagan and Feynman were so great. They could take complex problems and reduce them (read communicate) in a good way that most could understand and reason about without needing to traverse the entirety of the knowledge pyramid. Similarly, that is what Numberphile does so well.

x1798DE 10y ago

Interesting that this is entirely in English. Is this the English language version for foreign students, or are Japanese masters students expected to be fluent in English?

salesguy222 10y ago
Not sure about Japan but a very large number of computer scientists also speak English. Due to the youngness of the field, the most important material hasn't been translated from its source language, which is English :)
ldjb 10y ago
There is also a Japanese language version: http://www.i.u-tokyo.ac.jp/edu/entra/pdf/archive/16math_j.pdf http://www.i.u-tokyo.ac.jp/edu/entra/pdf/archive/16math_j.pd...
- lovich 10y ago
  That's interesting. I know Japanese has multiple scripts. Problems 1 and 3 appear to be in a much simpler script that looks less dense. Problem 2 is using whatever script that looks closer to Chinese and is full of much more complicated characters. Is there a reason for that? Did Japan import whatever concept question 2 is about from China and developed the math for the other questions independently?
  - sithadmin 10y ago
    As someone with moderate Japanese literacy, I don't really see the distinction between the problems that you're referencing. All problems here are using the same mix of phonetic and logographic characters, which is typical of modern written Japanese. >Did Japan import whatever concept question 2 is about from China and developed the math for the other questions independently? Very unlikely. Japanese concepts expressed by the borrowed Chinese logographic script (known locally as kanji) are frequently quite different than the meaning that would be expressed in Chinese. Additionally, choosing to write a text with relatively dense usage of kanji characters doesn't necessarily imply anything meaningful. For one, the swap between phonetic script and kanji often plays the role that the empty 'space' character plays in western languages. Sometimes the concepts being discussed include words that are homophones/homographs that are difficult to distinguish between in phonetic script, but are easy to distinguish with kanji. For instance, 'hashi' = chopsticks or bridge depending on emphasis. Aside from use of context it's not necessarily clear what you mean if you spell it out, but there are two distinct kanji for either 'chopsticks' or 'bridge' that are usually known, so it's easier to use those to ward off ambiguity. In other cases, choices between use of phonetic scripts or kanji simply comes down to nothing more than convention.

thearn4 10y ago

Some number theory, linear algebra, calculus of variations, combinatorics, etc. From the applied math side of things, it isn't terribly deep. It's actually right about what I would expect from an applied math student who was good at the undergraduate level and ready to study at a competitive university. For general admissions for any discipline though, I'll agre it's probably a bit rough for non applied math grads. At least it doesn't include any analysis, abstract algebra (groups, rings, fields, etc.), or topology. Which would be murder for non-math majors.

ajarmst 10y ago
Yep. My Math is undergraduate Electrical Engineering and Graduate Comp Sci. Some of the stuff I'd have to bone up on because I haven't done it in a few decades, but all the problems were familiar and seemed tractable---seemed appropriate for a recent maths undergraduate. (That said, I'd probably bounce hard off of the time limit, and it's been an awful long time since I did any maths where I wasn't allowed to pull a book down off a shelf.)
n00b101 10y ago
Honestly, I thought it was an undergrad entrance exam and was feeling sorry for Japanese high school students, before I checked the title again.
- tonyedgecombe 10y ago
  Yes, I made that mistake.
bloaf 10y ago
I missed the "graduate" part of "graduate school" and was temporarily impressed that Japanese high school students would have to learn eigenvalues/vectors. The second question about rotations seemed right out of an AP calculus class, until I saw the differential equations.
- startupdiscuss 10y ago
  > I missed the "graduate" part of "graduate school" and was temporarily impressed that Japanese high school students would have to learn eigenvalues/vectors. I had the same experience, followed by a "No wonder Sony produced such great products!" moment.

zulrah 10y ago

And I was angry at my uni,that my cs curriculum contained almost no maths. After reading this paper I realized that I am glad I didn't have to study this. Some calculus would have been useful though, as I am in a machine learning field now

yardie 10y ago
That's really shocking to hear. When I was in university CS was almost exclusively math. At the time that was the delineation between CS and computer engineering. One was mostly theoretical expressed through equations. The other was just practical application of EE and software development. And the finance college had the MIS program. They were using java.
- ldjb 10y ago
  In my personal experience studying undergraduate CS, apart from the dedicated maths modules, we did use mathematical notation for lambda expressions, regular expressions, finite automata, Turing machines and the like. We didn't cover calculus or probability though (I think you were expected to learn that in secondary school). I vaguely recall that we touched on matrices and eigenvalues/vectors. But essentially, we were only taught the maths that we needed to know for CS. And of course there were units that didn't use much maths at all, such as systems development and human-computer interaction.
- canistr 10y ago
  Your post is equally surprising given that for Waterloo, our undergrad ECE curriculums covered all this material. Arguably, we covered significantly more math (albeit in Electrical Engineering, but there was significant overlap with Computer) than what the CS students did.

wayn3 10y ago

This is really really simple btw, for a grad school entrance exam. Anyone who has successfully studied mathematics for a year should be able to score 100% on this exam.

kutkloon7 10y ago
I think most American students would score far below 100%.
stephancoral 10y ago
Wow you sure are smart why don't you write an email to the faculty of the highly ranked university that wrote this test to let them know how easy it is? Oh wait, you didn't even realize that this is just one of many tests you take on the examination day and you also must orally defend your undergrad thesis to a panel. But go on, continue telling us how smart you are
- wayn3 10y ago
  im not telling you how smart i am, im telling you how easy the test is. go on, be butthurt.
  - waynisarapist 10y ago
    Still being a cunt I see. I hope you get raped and no one believes you
    - deleted 10y ago
      [deleted]
jps359 10y ago
1st question is Linear Algebra 2nd question is Calc 2+Differential Equations 3rd question is probability/stats an analysis class would probably help as well I guess it's pretty feasible to take all of those in a year, especially if mathematics is your area of study. This wasn't the case in engineering in my experience. In any case, this wouldn't be so bad with a 4 year degree. Not really any harder than other PhD candidacy exams I've looked at

ninjin 10y ago

Goodness me, this brings back memories having taken the entrance exam back in 2009. I am glad that they have removed the "We do not guarantee the accuracy of the English translation of the original Japanese version of the exam" clause. It should be noted that this is only one "session" of a series of exams that you take in a single day. If I remember correctly there were two computer science sessions as well, followed by a presentation of your thesis in front of the faculty once your exams had been graded. After this you had to wait for a few weeks for the final result. It should be noted that I don't find these exams to be particularly difficult, the main difficult lies in the fact that the number of potential topics are drawn from the whole University of Tokyo undergraduate curriculum, so if you studied as an undergraduate at a different school or country -- like me -- you are at a significant disadvantage. What I did was to look at ten years or so of exams to get some statistics on what was likely to be on the exam, then locked myself in my room for the whole of January to cover portions that I had not studied before -- like network protocol specifics. In the end, it worked out for me, but I got somewhat lucky in that there were two statistics questions that I could breeze through to cover for my weaker calculus skills given my old school's focus on discrete mathematics.

microcolonel 10y ago
Problem 1.1 seems to be within the realm of casual curiosity for somebody who hasn't even entered undergraduate school. It's basically "I know what a matrix is, and how a matrix multiplies with a vector".
- ninjin 10y ago
  The general structure of the problems is that they start out with easy questions and get progressively more difficult. If you don't get a single "last question" right, the committee will most likely tear you to pieces. At least this was the common understanding back when I was at Todai. Fun random fact, back in 2009 Todai had their own "brand" of toilet paper made from recycled paper -- old archived exams that is. I think that I still have a roll stashed away in a box somewhere. Using it did give you a bit of a weird feeling knowing the anguish old students must have gone through.

ajarmst 10y ago

Hah! I gave 3.1 - 3.4 as an assignment to second-year CompSci students literally three weeks ago. Although my version was to determine the values experimentally for n<=8 bits and then estimate values for n up to 64. Bonus marks awarded for a mathematical proof of the pattern (which isn't really all that bad if you apply probability to patterns of two bits).

danielweber 10y ago
What form does the answer to 3.3 look like? Would it break apart the case for abs(r-s) > 1, which is 0?

tasey 10y ago

Wow, people saying that it is "really simple". I feel dumb.

Insanity 10y ago
I think that if you have just passed your undergrad degree, these questions are indeed not _that_ hard, because the stuff is still quite fresh in your mind. If you have left university for some time, chances are you did not do any math in that time inbetween, if you are a software engineer. Software engineering requires logical thinking in a different way than mathematics, so we do not really keep our mathematics skills up to date. Looking at document, I can remember touching the necessary mathematics to solve this, but not having done any mathematics of the kind makes me too rusty in mathematics to be able to solve it. With a bit of revision, someone having seen this somewhere in undergrad will probably be able to solve it.
- ldjb 10y ago
  An undergraduate degree in maths, sure. But if your degree is in computer science, I think it's likely you'd never come across any of this during the course. Some universities may teach it, but there are probably plenty that don't.
  - whorleater 10y ago
    Any self respecting university that teaches a formal computer science degree would teach this level of mathematics. Mine went beyond and happily forced students to learn introductory quantum mechanics as well, which annoyed the CS students to no end.
    - icebraining 10y ago
      I agree; my undergraduate course (not in the US) wasn't even called Computer Science, it was "Software Engineering", but we definitively learned enough math to pass that test easily after the second year.
    - 10y ago

MariuszGalus 10y ago

Question 1. Linear Algebra Level Question 2. Calculus II Level Question 3. Probability & Statistics Level You learn how to solve all of this in Undergraduate American Engineering courses.

RodericDay 10y ago
Yep. I was going to say the same thing, seems like a pretty solid test. Series and linear algebra, calculus, probability sounds like just about everything I saw in my Mechanical Engineering degree.

tokyoooo 10y ago

In problem 2, the Euler Lagrange equation imply that we are looking for the function y(x) such as the surface is minimal, so the solution is a cylinder, ie y(x)=constant=c=2.

piefoot 10y ago
The solution to the Euler equation gives the minimal surface with is obtained when y(x) is a catenoid, y=c cosh(x/c) here c satisfies 2=c cosh(1/c). The Euler equation gives the minimal distance when the function F is the length of the arc.

ajeet_dhaliwal 10y ago

I believe I could have done all of these questions 12-16 years ago. Shockingly I can't do any of them anymore without refreshing myself for what they are even asking in some cases.

deleted 10y ago
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kuon 10y ago

I would have been able to do this with no effort 20 years ago, but nowadays I would be totally incapable of solving even one of the problem. Damn I feel the years.

tluyben2 10y ago
Same feeling here. I only kept up with discrete math and the rest seems completely lost...

_-__--- 10y ago

I get stuck after the first few subproblems on all three. Does anyone have a PDF of the solutions? Are the solutions ever released? I'm a little rusty on my linear algebra.

ldjb 10y ago
Unfortunately, the page the PDF is linked from [0] does not provide solutions. I don't think they are available (officially, at least). [0] http://www.i.u-tokyo.ac.jp/edu/entra/examarchive_e.shtml http://www.i.u-tokyo.ac.jp/edu/entra/examarchive_e.shtml
Smaug123 10y ago
I whipped up a quick answer to q2; I don't know if it's right. https://www.patrickstevens.co.uk/misc/TokyoEntrance2016/TokyoEntrance2016.pdf https://www.patrickstevens.co.uk/misc/TokyoEntrance2016/Toky...

rotskoff 10y ago

No algebra at all---seems very odd because much of modern mathematics relies heavily on group and field theory.

kbart 10y ago
Maybe they teach advanced algebra during graduate studies, so you are not expected to know it at the entrance exams? At least in my experience that was the case -- I've only learned group and field theory while doing my CS master.
- ninjin 10y ago
  Todai has a strong traditional science profile and traditionally science deals mainly with calculus and statistics. This leads to less of an emphasis on algebra for these exams since they are based on the undergraduate curriculum. It is regrettable, especially for those like myself that came from a school that emphasised algebra and theoretical computer science and only taught advanced calculus at a graduate level.
sidusknight 10y ago
This is applied math. There isn't any analysis, topology, etc, either.

m23khan 10y ago

Assuming it is directed towards admission into a Graduate Mathematics / Engineering Program, it looks pretty average to me. For a student who is in-form (e.g. a fresh graduate or a mature student who has taken 2-3 advanced calculus and algebra+stats courses as refresher), they should be able to perform.

0xfaded 10y ago

I'm a CS grad who is fairly strong with maths but never really did learn to write a proper proof. All of these questions I would have been able to "solve", but the proofs would have been very casual. What would be a valid answer it 2.3 where it basically asks you to integrate an expression of F after substituting y'=dx/dy?

Smaug123 10y ago
"2.3" is a bit ambiguous as a label; do you mean part 3 of question 2, or the part of question 2 in which equation 2.3 occurs? If you mean part 2 of question 2: my answer would be "dF/dx = del F/del y dy/dx + del F/del y' dy'/dx; but we know del F/del y = d/dx(del F/del y'), so dF/dx = d/dx(del F/del y') dy/dx + del F/del y' dy'/dx, which by the product rule is d/dx(y' del F/del y'). Now integrate both sides." If you mean part 3 of question 2: just substitute F(y, y') from (2.2) into (2.4).
- mwane 10y ago
  What is the difference between d and del? > but we know del F/del y = d/dx(del F/del y') ????
  - Smaug123 10y ago
    del refers to the partial derivative, with LaTeX symbol \partial. d refers to the total derivative. "del F/del y = d/dx(del F/del y')" is just the Euler-Lagrange equation.
    - mwane 10y ago
      Which means there is no real difference; if all the types are known, it's just some form of Hungarian notation?
      - piefoot 10y ago
        if a function depends only on multiple variables then those are partial derivatives, if the function depends only on one variable then there is only one derivative d/dx

hokkos 10y ago

What exam is it relative to that : https://en.wikipedia.org/wiki/Education_in_Japan#School_grades https://en.wikipedia.org/wiki/Education_in_Japan#School_grad... ? What are the age of the students ?

ldjb 10y ago
It's the entrance exam for graduate school, so candidates would likely be 21+.
TorKlingberg 10y ago
This is an entrance exam for graduate school, so around age 22–23.
- hokkos 10y ago
  Wow seems really easy for that age.
  - doall 10y ago
    Yes, the graduate school paper exam of Todai is not difficult compared to the undergraduate entrance exam for that age. The graduate one only requires the very basics to do research. Requirements are different.

dysoco 10y ago

Does anyone have links to Undergraduate entrance exams from other Universities? I am currently taking one in my country and would like to compare the level with mine.

ldjb 10y ago
I don't know about other countries, but here in the UK, most universities don't require an entrance exam to study computer science (they go by the grades you leave secondary school / sixth form with). The only two universities I'm aware of that have entrance exams for CompSci are Oxford and Cambridge (because they have to be different): Oxford: http://www.cs.ox.ac.uk/admissions/undergraduate/how_to_apply/MAT_test.html http://www.cs.ox.ac.uk/admissions/undergraduate/how_to_apply... Cambridge: https://www.cl.cam.ac.uk/admissions/undergraduate/admissions-test/ https://www.cl.cam.ac.uk/admissions/undergraduate/admissions...
piymis 10y ago
For India, we have entrance exams for admissions into IIT's. Here (http://cms.fiitjee.co/Resources/DownloadCentre/Document_Pdf_42.pdf http://cms.fiitjee.co/Resources/DownloadCentre/Document_Pdf_...) is one solved maths paper of it. Mind you, this is for admission in undergraduate courses straight out of school.

youareanelitist 10y ago

The elitist cunts that are in this chat are the reason why science will never matter to the world. But hey, at least you are better than everyone else right?

Daishiman 10y ago

For a grad school entrance exam this isn't too difficult. It's first and second-year maths stuff for a decent CS degree that doesn't skip out on the basics.

mrcactu5 10y ago

this is graduate school of information science, not the mathematics department of University of Tokyo here's the exam (in Japanese) http://www.ms.u-tokyo.ac.jp/kyoumu/documents/26english.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/documents/26english.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/docs/26a%200119.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/docs/26a%200119.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/docs/26b%200119.pdf http://www.ms.u-tokyo.ac.jp/kyoumu/docs/26b%200119.pdf

nscalf 10y ago

So I never really got too deep into math, once I decided I wasn't going into physics as a career I backed off, but this throws me back into 12th grade AP Physics exams. I'll never forget refusing to take a bad grade on a matrix problem(?) and skipping the next few classes to spend hours trying to do it the long way... I didn't get credit for that question. That being said, now that I'm on the other side of college and in a steady career, I'm really interested in diving deeper into math. Does anyone have any recommendations on resources for learning from Calculus on? I could probably do with a Calc refresher as well. Thanks!

jimsojim 10y ago
here's a post from mathstackexchange that I always get back to: http://math.stackexchange.com/questions/843697/learning-higher-mathematics-on-your-own http://math.stackexchange.com/questions/843697/learning-high...

khana 10y ago

Damn. I guess I ain't goin' to grad school.

nmca 10y ago

Interesting to see how it compares to my undergrad efforts: http://s000.tinyupload.com/index.php?file_id=00312832089149851850 http://s000.tinyupload.com/index.php?file_id=003128320891498...

piefoot 10y ago

Solving 3.1 I think the solution is binomial(n1+r-1,r-1) where n1=n-r, but I obtained that formula by calculating the formula for r=1, then r=2, and the relation f(r,n) = sum(f(r-1,n-i),i,0,n) and using maxima with simpsum and factor to obtain a simplified expression whose form suggest the general rule, but I don't see a simpler way.

Smaug123 10y ago
If you want something to Google, the result is completely standard, and it goes by the name of "stars and bars".

ejanus 10y ago

I have not truly figured out the difference between permutations and combinations. I know about permutations having 'order' but when I see real questions I got lost. Could someone help this 'lost' soul?