**FAQ - Frequently Asked Questions**

**Me and my webpage
**Who are you?

Why are your papers so boring?

What is the copyright status of your papers?

What software do you use to draw your graphics?

What are PS files and what am I supposed to do with them?

Tell me of some good online sources for elementary geometry.

Tell me of some good books on geometry.

Tell me of some good online sources for inequalities.

Tell me of some good books on inequalities.

Why are you not active on the MathLinks forum anymore?

Please give me the IMO Shortlist [insert year of last IMO here]!

Can you solve the following problem for me?

**Who are you?**

From 2006 till 2011, I have been studying mathematics and computer
science at the Ludwig
Maximilian University in Munich. I have
finished my diploma thesis in
July 2011.

Since September 2011, I am a graduate student at MIT,
Cambridge (MA), advised by
Alexander Postnikov.

I was
born in Moscow in 1988 and have moved to Germany in 1996. I have
spent my school years in Münster and Karlsruhe, Germany,
participating in mathematical olympiads like the Bundeswettbewerb
Mathematik, the Deutsche Mathematik-Olympiade (both German
mathematical competitions) and the International Mathematical
Olympiad.

Have won first prizes in the 1st Rounds of the Bundeswettbewerb
Mathematik 2001-2006, as well as in the 2nd Rounds of the
Bundeswettbewerb Mathematik 2003-2005 (where I gained third
prizes in 2000 and 2002), and became Bundessieger 2003, 2004, 2005 and
2006. Participated in the Deutsche Mathematik-Olympiade since
2002, and gained a 1st prize in the 41th olympiad 2002 (Hamburg),
an honorable mention in the 42th olympiad 2003 (Bremen), a 2nd
prize in the 43th olympiad 2004 (Essen) and 1st prizes in the
44th olympiad 2005 (Saarbrücken) and the 45th olympiad 2006
(München). Silver medals in the 45th IMO (International
Mathematical Olympiad) 2004 and in the 46th IMO 2005, and gold
medal in the 47th IMO 2006.

My current field of interest is algebra, especially constructive
algebra, representation theory and
combinatorial algebra (or algebraic combinatorics; this includes Hopf
algebras). I occasionally
still spend some time
on olympiad mathematics (mostly combinatorics).

**Why are your
papers so boring?**

I guess this is either due to the subject or to the length of my
papers.

As for the subject: This website (particularly its geometry part)
contains a lot of stuff I have
written since 2002 (when I was 14). Some of it is
pretty much junk, some was posted just in order to answer some
people's questions, some other notes were written out of boredom...
don't expect to find anything sensational here.

As for the length: I am trying to keep my proofs as
detailed as possible - not only in order to make them
understandable to less experienced readers (as far as this is
possible - some of the proofs still assume a rather advanced
background),
but also to make maximally sure they have no flaws - mistakes
often hide in "trivial" or "left to the
reader" clauses. As a consequence, a
proof that takes 1 page in someone else's writeup often
takes 4 pages in mine... But let me hope that these 4 pages won't take
4 times as much time to read as someone else's 1 page! In fact, if you are
experienced in the respective fields, you might be able to
speed-read much of my notes anyway.

What also boosts the length of my geometry notes are the many big
graphics. I believe that in times of harddisks with terabytes of
storage space, some additional 30KB coming from more (vector!)
graphics in a geometrical text are not much of a waste.

**What is the
copyright status of your papers?**

Every file which is

**1.** hosted on this webpage -- i.e., its URL begins with
"http://web.mit.edu/~darij/www/" or with
"http://www.cip.ifi.lmu.de/~grinberg/" --, **and**

**2.** is hyperlinked from an .html (or .htm) file on this webpage,
**and**

**3.** the word "[Prop]" does not appear right in front of the
hyperlink that links to it

is **public domain**, i. e. you can
use it in whatever way you want without asking me for it.

This is not an invitation to plagiarize, but
plagiarism is not about copyright anyway. The reason for condition
**2.** above is to exclude "secret"
files which are (usually temporarily) hosted on my website (e.g.
to exchange them between my different machines). Condition
**3.** is mainly meant to deal with joint papers which have
not been fully written by myself. I am a friend of GPL, LGPL,
CC and the likes, but I think they are overkill when applied to
mathematical writing.

**My posts** on the MathLinks forum, the PEN forum, the Hyacinthos newsgroup and
in other internet resources are **public domain as well**,

**with the exception of:**

- quoted text (text inside "Quote" tags);

- attachments to posts (some of these are public domain, some
aren't);

- posts on MathLinks beginning with the sentence "The
author of this posting is:" (in fact, these are posts made
by others which were lost during a hacker attack on MathLinks,
and which I reposted because I happened to have backups of them).

**What software do you use to
draw your graphics?**

I got asked this several times. Unfortunately, the answer is
unlikely to help anyone.

(Note - As of 2011, this answer won't
be of help even to myself, as I don't have MS Word anymore. But as I am
not doing geometry anymore, I don't care.)

First, my notes are written with Scientific
Workplace (SWP) - a (commercial, and,
for its quality, way overpriced) graphical LaTeX user interface.
It generates somewhat messy LaTeX code, with a few additional
features - the one I use is the ability to include WMF vector
graphics.

The graphics are drawn with the dynamic geometry software Euklid DynaGeo by Roland Mechling. As far as I know, this software is
distributed in German only (and being written for school use
mainly, it is not even the best choice). Then, through the
Windows clipboard, these graphics are either directly exported
into SWP as WMF, or first edited in MS Word and then exported
into SWP as WMF.

As you could have guessed, this way of handling graphics is very
prone to bugs, and I would not recommend this to anyone, even in
the unlikely case you have both SWP and Euklid DynaGeo installed
on your computer. I would sincerely like to know a better way to
get geometric sketches embedded into TeX files in a vector format
(please don't tell me about bitmap-based solutions, I don't like
pixeled lines).

**What software do you use to
draw your commutative diagrams?**

I just use the (rather standard) Xy-pic package. See a
short introduction on wikibooks and J. S. Milne's tutorial.

If you find its syntax somewhat too spartan, you can try also try out
Michael Barr's diagxy
package, which provides a kind of UI for the \xymatrix command.

See also J. S.
Milne's page for a
list of other diagram packages.

**What are PS files and
what am I supposed to do with them?**

Some papers on this site are downloadable in PS (= PostScript)
format only. These are notes I have written some years ago when I
wasn't able to convert from TeX to PDF yet. You can **view
PS files under Windows** by installing the following two
programs (in this order): **-- Being not experienced, I can
only hope that the below instructions will work. --**

**Ghostscript.**You need either GPL Ghostscript or AFPL Ghostscript.

For**GPL Ghostscript**, go to http://sourceforge.net/project/showfiles.php?group_id=1897, scroll down to "GPL Ghostscript" and download the file ending with "w32.exe" (supposing you have a 32-bit version of Windows) corresponding to the latest avaliable GPL Ghostscript version.

For**AFPL Ghostscript**, just download it off http://pages.cs.wisc.edu/~ghost/doc/AFPL/.

I don't have enough experience to tell you which of these is better. AFPL Ghostscript is not developped further anymore since 2006 and is not "as open source" as GPL Ghostscript, but it is easier to obtain.**GSView**or, alternatively,**SumatraPDF**(allow Javascript to use the download page). Either of these is sufficient, but it has to be installed**after**you have installed Ghostscript.

You won't ever have to run Ghostscript (it just needs to be
installed); you
will open PS files in either GSView or SumatraPDF.

Note that SumatraPDF can also read most PDF files (I haven't seen
a mathematical paper that SumatraPDF couldn't read) and is pretty
fast at that
(I personally use SumatraPDF as the default viewer for both PDF
and PS). GSView can also read PDF, but it misses a lot of features
like antialiasing.

**What is the QEDMO?**

QEDMO stands for QED Mathematical Olympiad. The QED is an organization of German mathematical olympiad
participants based in Bavaria who organize meetings and seminars
for each other. From late 2005 on, some of these meetings feature
a math fight (an oral mathematical contest, with two teams
solving questions and debating the solutions in front of the
blackboard) called the QEDMO. The problems are of varying
difficulty (some very basic problems occur on every QEDMO, but a
few of the problems have the level of an IMO problem 3). These
math fights are usually organized by Daniel Harrer and me, with
problems partly taken from different sources, partly invented by
ourselves (Peter Scholze has also done some work here).

Until now (June 2007), four QEDMO's have been performed. Proposed
solutions for the number theory and combinatorics problems are
usually written by Daniel Harrer, and those for the geometry and
algebra problems are written by me. The former will be avaliable
as soon as the QEDMO gets an official website; the latter are,
upon completion, downloadable from the QEDMO section of my website.
However, all of these solutions are in German. If you are
searching for solutions in English, you can try searching for the
problems on the
MathLinks forum.

However, you can download the statements of the problems in English. I have tried to retrace the sources of the
non-original problems, but this turned out more difficult than I
expected.

**Tell me of
some good online sources for elementary geometry.**

Please visit the links
section of this website.

[I will be glad to know if you have good links that can be added
to this section!]

**Tell me of
some good books on geometry.**

I hope you mean *elementary* geometry (as opposed to
topology, differential or integral geometry which are absolutely
different fields of mathematics).

First, please visit the links
section
of
this
website
to find some online literature. It has the advantage that it is
free and doesn't take time to ship. You can learn a *lot*
from the Cut-the-Knot
website and from Kedlaya's
Geometry Unbound. Leites'
translations of books by Victor Prasolov are invaluable to a
problem solver who wants to get better in geometry.

Now, if you really want
books in the classical meaning of this word (as in: these things
printed on paper, offline, without hyperlinks...), the first choice is
obviously

H.
S.
M.
Coxeter,
Samuel
L.
Greitzer,
*Geometry Revisited*

(this one has been printed several times, and as far as I know,
the editions only differ in the solutions to the exercises - the
newer ones have partly better solutions). The first three
chapters of this book contain some real common knowledge on
triangle geometry and radical axes theory. However, the
"introduction into projective geometry" chapter is
anything but an introduction into projective geometry, and the
way inversion is introduced is far from complete.

The next textbook is

Nathan
Altshiller-Court,
*College Geometry*.

This has been reprinted in 2007 after having been elusive for a
long time. It contains nearly all plane geometry you might need
on contests.

Along with this book,

Roger
A.
Johnson,
*Advanced Euclidean Geometry*

has also been reprinted. While I would not recommend this for
olympiad training, it can be helpful for a more serious study
since it contains a number of lesser known results.

Then, there is

Ross
Honsberger,
*Episodes in Nineteenth and Twentieth Century
Euclidean Geometry*.

This one is really fun to read if you are into elementary
geometry. Much of the content can help you in mathematical
olympiads as well - though, for an IMO gold medaillist with focus
on geometry, this one will be rather like a collection of simple
exercises (which can still be of use).

I have now put up a page for errata in
these three books (some other books may also be included).

Unfortunately, this is pretty much all books I can recommend -
apart from the (also rather few) German and Russian ones I won't
itemize.

**Tell me of
some good online sources for inequalities.**

Here is some random stuff I found useful. In fact, I have never
learnt inequalities systematically in online sources - most of my
basic knowledge comes from the German IMO training, and the rest
is experience from solving MathLinks
problems and reading others' solutions.

Thomas Mildorf
has a
nice
script
on
inequalities. (A newer version used to be at Mildorf's
MIT home page, but that is now offline.)

Hojoo Lee is
rather known for his
"Topics in Inequalities".

Kiran Kedlaya has another
text similar to the two above.

This
MathLinks topic is partly of interest, and the whole Inequalities
Theorems&Formulas section has a number of interesting
discussions, along with tons of spam.

If you read German, Robert
Geretschläger has his
script as well.

Anyone knows more sources? Just tell me...

**Tell me
of some good books on inequalities.**

There are several books by the (Romanian) GIL publishing house
on olympiad-style inequalities. Two of them are in
English. Please don't ask me how to order books published by GIL
from outside Romania - apparently this has not been organized
smoothly yet :( . Let me make some words about these two books:

Vasile Cîrtoaje, *Algebraic Inequalities - Old and New
Methods*, Gil: Zalau 2006.

This one is 480 pages long and features many interesting tactics
and examples on solving inequalities. Unfortunately, you are not
likely to enjoy all these 480 pages, because many of the modern
methods for solving inequalities include applications of calculus
and involved computations. However, *a lot* was done to
keep these ugly parts at a minimum while keeping the whole power
of the new methods.

The RCF ("right convex function"), LCF (guess what this
means) and EV (equal variables) theorems as well as the AC
(arithmetic compensation) and GC (geometric compensation) methods
provide a means to solve >95% of olympiad inequalities using
rather straightforward - not nice, but doable - computations. All
of these methods are extensively presented with numerous
examples. A short chapter underlines applications of the
(underrated) generalized Popoviciu inequality. Finally, and - in
my opinion - most importantly, a lot of exercises with solutions
are given which don't require any strong new methods, but just
creative ideas and clever manipulations.

Pham Kim Hung, *Secrets in Inequalities (volume 1)*, Gil:
Zalau 2007.

This one has 256 pages, and is remarkable for mostly avoiding
computations. Numerous creative ideas can be found here - I was
particularly surprised about some of the applications of the
Chebyshev and rearrangement inequalities. Besides, a good
introduction into the applications of convexity is given. I would
recommend this book to olympiad participants who look for
challenging problems and intelligent techniques without the aim
to be able to kill every inequality.

**Why are you not active
on ****the
MathLinks
forum**** anymore?**

As of 2010, I have moved on to advanced mathematics. This doesn't mean
I am not doing elementary combinatorics once in a while, but I am not
that centered on olympiad problems anymore. You will more likely find
me on MathOverflow asking and
answering questions. I still do roam MathLinks from time to time,
mostly the Linear Algebra and Superior Algebra subforums.

If you are here for my PEN (Problems in Elementary Number Theory)
solutions, you can find the source code of my (relevant)
posts here.

**Please
delete/edit/move/merge some MathLinks post for me!**

Okay, give me a link to your post. However, please note that

- **I am never going to remove any mathematical content**
without a **very good** reason. When you post
something on MathLinks, you are giving it away to the community
and you should not be able to take it back again. (And experience
shows that people who want their mathematical posts deleted often
happen to be cheaters who want others to solve an olympiad
problem for them and then to have it deleted in order to cover
the tracks.)

- I am only a moderator in the Olympiad section (minus Inequalities),
the College
section and parts of the National Olympiads subforum. This means
that, among other things, I can move topics from the Olympiad
section into Pre-Olympiad, but I *cannot* move topics from
Pre-Olympiad into the Olympiad section.

**Please give me the
IMO Shortlist [insert year of last IMO here]!**

Chances are high that I don't even possess the Shortlist.

However, even if I do, I am not entitled to distribute it
further.

Most people who ask me about shortlists do know pretty well why
these are kept confidential until the next IMO. For anyone else:
The Shortlist of year *t* contains the problems that were
"almost" selected for the IMO of year *t*. There
are *y* (usually *y* is approximately equal to 30)
such problems, and 6 of them are finally selected as the IMO
problems. The other *y* - 6 problems are used by various
countries in their team selection tests for the IMO of year *t*
+ 1. Therefore, publishing these problems can make team selection
in other countries unfair, as some participants will know the
selection problems in advance from the internet.

Of course, nowadays everybody discusses everything online, and
there are various places where IMO Shortlist problems leak into
public access. I, among other training participants, found it fun
to hunt down such leaked problems through the internet in my IMO
training times - but we were "fair" enough to take them
from publically accessible websites only (rather than asking
people for them via mail) and to disclose our knowledge to the
team leaders (what did not stop us from intentionally creating
havoc by doing this as late as 1-2 days before the actual test,
so that the whole test had to be rewritten in these 1-2 days...
so much for actual fairness). If you are into this kind of games,
nothing keeps you from doing your own googling but I won't
provide you any undisclosed data.

Wait until IMO *t* + 1 to find the IMO Shortlist *t*
freely accessible online, most likely with solutions.

**Can you solve the
following problem for me?**

I sometimes get emails and other messages from people
asking me to solve mathematical problems.

Unfortunately, I mostly do not have the time to reply to such
messages. If you want a reply, please keep the following things in mind:

- Please post your problem **on ****the
MathLinks/AoPS forum** (and then send me, by email, a
link to the post). This serves three purposes: First, even if I don't reply, somebody else may.
Second, the solution (or hints) will be of much more use when posted
publicly on the internet. Third, it is the only way I can really be
sure that you are not trying to cheat on a contest by letting me solve
the problems for you (sorry, but people do try to cheat this way). (Exception:
If you want to collaborate with me in writing a research paper, then
you can send me the problem via email.)

- My current fields of expertise are linear
algebra, Hopf algebras and representation theory. Also I have
some knowledge of commutative
algebra, combinatorics (enumerative, extremal and graphs; no games) and olympiad-level number theory. I
used to be good at elementary geometry (in the plane) and inequalities,
but nowadays am not interested in these anymore (exception:
inequalities with a combinatorial flavor, e. g. majorization theory). I
have pretty much no experience with combinatorial geometry, functional
equations, functional analysis, complex analysis, topology (except for
algebraic) etc.

FAQ - Frequently Asked Questions

*Darij Grinberg*