
University of Tokyo Graduate School Entrance Exam – Mathematics (2016) [pdf] - v3gas
http://www.i.u-tokyo.ac.jp/edu/entra/pdf/archive/16math_e.pdf
======
ninjin
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
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".

~~~
ClassyJacket
Only vaguely related, but I've made it all the way through high school (doing
the 2 hardest of three maths subjects) and a software development degree
without ever having a lesson on Matrices.

There really seems to be a hole in the system somewhere.

~~~
microcolonel
Yeah, matrices are also very useful. Instead of needing the "Tribonacci"
equation definition, you can approximate or exactly match a large number of
equations and operations with matrices.

Anyway, there are many things missing. I feel that _An Introduction to Law and
Legal Reasoning_ should be standard middle-school reading; but alas, civics
has not been part of the curriculum in my life time.

------
ldjb
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....](http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-s.pdf)

February 2016:
[http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-w....](http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/2016computer-w.pdf)

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.pd...](http://www.i.u-tokyo.ac.jp/edu/course/cs/pdf/H29csguide_e.pdf)

~~~
v3gas
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
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!).

------
demonshalo
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!

~~~
wayn3
Mathematical language is precise. Problem 1 of this problem sheet is expressed
in precisely as many words and symbols as is required.

You dont need any more language to fully understand the problem. Any less, and
the problem would not be correctly specified.

That is the beauty of mathematics. The things you say are always falsifiable.

When taking a math exam, you can not contest having "misunderstood the
question". The question is precise. By saying you "misunderstood it", you
acknowledge that you don't understand the topic well enough to pass the exam.
There are no ambiguities in correctly expressed math.

Whatever youre going to write down as your answer for problem 1a) is either
correct or its not. No amount of arguing is going to change that. We don't
need to write essays and we don't want to. We just apply what's been proven
correct to solve new problems.

If math problems were "communicated in more natural ways", they would be
reduced to problems people are going to argue and reason about, like they do
in a liberal arts class. If you don't want cough up the required rigor, just
go do something different.

Mathematics does not make amendments for the rigor impaired. If you want to
study something math-y without all the rigorous proofs, physics is what you're
looking for.

~~~
demonshalo
> When taking a math exam, you can not contest having "misunderstood the
> question". The question is precise. By saying you "misunderstood it", you
> acknowledge that you don't understand the topic well enough to pass the
> exam.

This right here is completely unfair and infuriating.

Great mathematicians always find themselves taking the wrong turn and
producing the wrong result(s) because they "misunderstood the question".
Granted they learn form their mistake and revise and iterate until they get
their theories and proofs right. But if they constantly fail, how could you
then claim that they "don't understand the topic well enough". Are you telling
me that Tao does not understand prime numbers well enough because he cannot
solve for formula X?

This is the 3rd post that I reply to that seem to illustrate my point. You are
RIGHT about the nature of mathematical language. But you CANNOT appeal to the
general public with that arrogant attitude. You cannot communicate using dense
language and expect non-math people to relate or even care. Granted, you might
not give a fuck about that. But that is counter productive because if you want
more rigor in society you should introduce more people to math rather than
having them being put-off by some elitist attitude.

Once again, this is a M.SC. Exam so it should be this way. My issue is with
math communication to the public in general, not this pdf specifically!

~~~
wayn3
I don't expect non-math people to relate or even care. If you open a textbook
on algebraic topology and you've never studied even the most mundane calculus,
you won't understand anything.

Somehow, this infuriates people.

The same people would have no problem admitting that they should probably not
read specialized literature on law that requires readers to have gone through
8 years of law school before they can make sense of the content.

Mathematical literature is not meant for casuals. Its impossible to express
advanced concepts concisely, in such a way that people who don't know the
material can understand them.

This is like asking someone for an explanation on how to walk and then
throwing a fit when they reply with "well, place one foot on front of the
other in such a way that you dont fall over". You don't expect them to discuss
activation levels in skeletal musculature resulting from biochemical processes
in the nervous system. But in Math, thats what you want. Clear and concise
language that inspires and is fun AND fully describes the problem at the same
time.

Not going to happen.

~~~
demonshalo
You still don't get it... You are arguing against something I did not say!

~~~
wayn3
Theres no elitism in math. We just require you to write proof in such a way
that it is actually a proof, and not just some idea you've had.

In order to do that, you need to study math. Not because we are elitists but
because you need to learn how to prove something. You need to learn the tools.

The same way a mason expects you to learn his craft before he takes you
seriously. If you don't have a clue about masonry, you can still semi reliably
build a brickwall in your backyard. But not a cathedral. Obviously, you dont
expect anyone to allow you to work on a cathedral. But you expect
mathematicians to take your thoughts on math seriously.

Its like a bootcamp idiot trying to piss all over a big software project. Just
because they know how to vomit on a keyboard doesnt mean they get to make
decisions.

~~~
demonshalo
you STILL dont get it!

------
thearn4
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.

~~~
n00b101
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
Yes, I made that mistake.

------
MariuszGalus
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
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.

------
ajeet_dhaliwal
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.

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

~~~
Insanity
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
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
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.

~~~
sidusknight
MIT, for example, doesn't require their CS students to take a differential
equations class. So you're wrong in that respect. The rest is certainly doable
though.

~~~
whorleater
There's nothing in this exam that requires specific diff eq knowledge.

~~~
sidusknight
Doesn't question 2, problem 2?

~~~
whorleater
No? That's calc 2 level problem, not a diff eq one. Partial derivatives are
hardly diff eq territory.

~~~
sidusknight
Sorry, I'm not american. What does calc 2 normally refer to in the US?

------
mrcactu5
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)

------
m23khan
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.

------
zulrah
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
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
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.

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

~~~
kbart
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
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.

------
nscalf
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
here's a post from mathstackexchange that I always get back to:
[http://math.stackexchange.com/questions/843697/learning-
high...](http://math.stackexchange.com/questions/843697/learning-higher-
mathematics-on-your-own)

------
Daishiman
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.

------
x1798DE
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?

~~~
ldjb
There is also a Japanese language version:

[http://www.i.u-tokyo.ac.jp/edu/entra/pdf/archive/16math_j.pd...](http://www.i.u-tokyo.ac.jp/edu/entra/pdf/archive/16math_j.pdf)

~~~
lovich
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
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.

~~~
glandium
> For instance, 'hashi' = chopsticks or bridge depending on emphasis.

or edge.

For the curious readers, since you didn't include the corresponding chinese
characters:

hashi (chopsticks): 箸

hashi (bridge): 橋

hashi (edge): 端

------
wayn3
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.

~~~
jps359
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

~~~
kutkloon7
Oh wait, is it for a PhD program? Then it is pretty easy indeed. Although I
would still possibly choke on proving that the surface area equals that
formula.

I mean, are you supposed to use the standard formula for surface of
revolutions
([https://en.wikipedia.org/wiki/Surface_of_revolution#Area_for...](https://en.wikipedia.org/wiki/Surface_of_revolution#Area_formula))?
Are you supposed to derive this formula yourself?

Strictly speaking, I don't even know the definition of surface.

------
kuon
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
Same feeling here. I only kept up with discrete math and the rest seems
completely lost...

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

------
piefoot
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
If you want something to Google, the result is completely standard, and it
goes by the name of "stars and bars".

------
0xfaded
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
"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
I think a major problem with mathematical language is lack of discoverability.
If I don't understand a specific symbol or notation, there is no way to click
"go to definition" or look it up on Google.

Furthermore while mathematicians pride themselves on rigor, I would bet that a
lot of proofs require the reader to fill in incomplete steps and/or contain
notational mistakes. Just try to imagine what kind of code you would end up
with if you did not have access to a compiler and relied only on your logical
reasoning for writing correct code. Even if the code was reviewed by other
peers, who themselves do not have access to a compiler, they would miss some
mistakes because, being familiar with the subject, their brain would fill in
the correct meanings.

~~~
Smaug123
I think this might be attached to the wrong comment?

~~~
mwane
Not really. It's simply a general observation I made after reading your post.

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

~~~
TorKlingberg
This is an entrance exam for graduate school, so around age 22–23.

~~~
hokkos
Wow seems really easy for that age.

~~~
doall
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.

------
nmca
Interesting to see how it compares to my undergrad efforts:
[http://s000.tinyupload.com/index.php?file_id=003128320891498...](http://s000.tinyupload.com/index.php?file_id=00312832089149851850)

------
ejanus
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?

------
tokyoooo
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
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.

------
_-__---
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.

~~~
Smaug123
I whipped up a quick answer to q2; I don't know if it's right.

[https://www.patrickstevens.co.uk/misc/TokyoEntrance2016/Toky...](https://www.patrickstevens.co.uk/misc/TokyoEntrance2016/TokyoEntrance2016.pdf)

------
dysoco
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.

~~~
piymis
For India, we have entrance exams for admissions into IIT's. Here
([http://cms.fiitjee.co/Resources/DownloadCentre/Document_Pdf_...](http://cms.fiitjee.co/Resources/DownloadCentre/Document_Pdf_42.pdf))
is one solved maths paper of it. Mind you, this is for admission in
undergraduate courses straight out of school.

------
youareanelitist
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?

