Thoughts on Construction

Ankan Bhattacharya
Last updated October 30, 2021

Contents
1 What is this? 2

2 Introductory thoughts 2
2.1 Thanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Disclaimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 On contest criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.4 Problems discussed in this document . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Problem construction 101 5

3.1 So, how do you do it? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Working forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Working backwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.4 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 General guidelines for problem creation 7

4.1 Make the problem look interesting . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 On answer extractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 On contrived problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.4 On computational problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.4.1 On computational answer extractions . . . . . . . . . . . . . . . . . . . . 10
4.4.2 Specific tips for computational problems . . . . . . . . . . . . . . . . . . . 11
4.5 Never settle! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.6 Some possible pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.6.1 On functional equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.6.2 On configuration geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.6.3 On olympiad-style computational inequalities . . . . . . . . . . . . . . . . 13

5 General guidelines for contest creation 14

5.1 Format your problems nicely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Formatting tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.1 On math formatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.2 On typesetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.2.3 On English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3 On official solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.4 On releasing problems not used for contest . . . . . . . . . . . . . . . . . . . . . . 15

6 Hall of fame 16

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§1 What is this?
I’m not really sure. It was originally intended to be a document on “how to create math contest
problems,” but that’s not an easy topic to write an article about. So here’s this instead.

§2 Introductory thoughts
§2.1 Thanks
Thanks to certain people for discussing problem quality philosophy with me. You know who you
are. In particular, I’d like to thank David Altizio for his wise words of wisdom.
Thanks to Evan Chen for creating evan.sty, the style file used to typeset this document.
In general, I’d like to acknowledge Evan for all that he’s done for the math contest community,
and for me in particular. In late 2017, Evan offered me an opportunity to submit problems for
and review USA TST(ST) packets, and in late 2018, offered me an opportunity to be a USA
TST(ST) coordinator. It’s pretty safe to say that my life would be completely different if these
two events hadn’t taken place.
I would also like to commend Evan for being very transparent with contest processes. Whenever
I suggested problems for any contest which Evan was involved with, I was confident that my
problem materials would be kept secure, and that I would always know the status of my proposals.
I have seen other proposers (including Evan himself) “lose” potential problem proposals because
their proposals were not managed appropriately.
Above all, I’m really grateful to be involved in math contests that respect their problem
proposers.

§2.2 Disclaimers
I use various contest problems as examples for this document, and not all of the problems are
portrayed positively. These criticisms are intended to be criticisms of the problems themselves,
not of their authors. (I make plenty of problems that I don’t consider good. The public just
hasn’t seen most of them.)
Problem quality is an inherently subjective thing, and as such, this document will be influenced
by my opinions. Most of them are fairly mild, but my beliefs on geometry are somewhat extreme.
You have been warned.
In this document, I assume that problems and contests are being developed in English. If you
are developing in another language, feel free to make the corresponding substitutions.

§2.3 On contest criticisms

Throughout many years of being involved in the math contest community, I’ve seen many
criticisms of specific math contest instances. Usually, most of these criticisms are quickly shot
down by others in the community (especially for instances of well-respected contests), with one
of the most common counters being the following.

Well, making contest problems is hard. Why don’t you try it yourself?

(Sometimes, the first sentence is removed.)

This counter makes me feel uneasy. What will a contestant gain from constructing problems?
They don’t have the opportunity to propose problems to the contests they’re participated in.
The truth is that most review processes for contests (at least for contests I’ve helped review)
kinda suck: there are far too many problems to thoroughly examine each one. (Shoutout to the
IMO PSC, which processes about 150 problems each year. I have no idea how they do it.)

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Here are some horror stories. The purpose of giving these stories isn’t to suggest “Wow, Ankan
must be so smart: he found two broken problems, and almost found a third.” Rather, it’s the
opposite: I believe reviewers don’t spend enough time thinking about problems and how they
can be broken. This is not something I can really blame reviewers for: most people have lives.

USA TSTST 2018/3 This problem did not initially have the ∠A 6= 60◦ condition, which
renders the problem false. This issue went unnoticed throughout the entirety of TSTST packet
review. Eventually, during June 2018 (about one week before MOP 2018 was due to start), I
noticed the issue, and only because Evan suggested I proofread the problem.

USEMO 2019/6 This was not a very popular problem, and for good reason: the problem is
completely cooked, in the sense that X, Y , and Z can be any points on the altitudes. This was
not noticed by any reviewers, possibly because the problem was already quite unpleasant. I had
the thought to check this, but selfishly assumed that if I had the thought, so would someone else,
and surely they’d check it. (This USEMO also did not have a testsolving phase, which might
have contributed to no one noticing the cook.)

USA TST 2021/2 This problem did not initially involve the length inequality conditions,
without which the problem is false: the fixed point could be at infinity. This problem had been
proposed to multiple contests, and as far as I am aware, this issue was never noticed in the
review processes of any of those contests. I only noticed it when we were assembling the TST.

§2.4 Problems discussed in this document

Apart from the above three problems, the following problems are discussed in the main document
(in the order shown below), and many of them are spoiled rotten. You have been warned.

1. OMO Spring 2019/23

2. USA TST 2019/6

3. RMM 2021/5

4. USEMO 2020/5

5. USA TSTST 2019/9

6. IMO 2018/6

7. Shortlist 2019 N5

8. IMO 2013/6

9. AIME 2014/I.14

10. AMC 2015/8.25

11. USEMO 2019/5

12. Erdös-Szekeres theorem

13. USAMO 2018/4

14. Taiwan TST 2015 (on nine-point circle of 4BIC)

15. Shortlist 2019 G6

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16. Shortlist 2020 G6

17. Korea Finals 2019/2

18. IMO 2006/3

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§3 Problem construction 101

This section aims to answer the question “How do you write problems?” For “How do you write
problems well,” see the next section.

§3.1 So, how do you do it?

There’s no real science to it. If there was, I’d be following it, and I’d have at least 64 published
olympiad problems by now. If you’re frantically reading this because you promised a contest by
tomorrow, don’t have problems for it, and are hoping I’ll magically tell you how to make them,
I can’t really help you. Sorry. Maybe give yourself more time next attempt?
With that out of the way, there are two broad philosophies for constructing problems:
constructing forwards and constructing backwards. (If you’ve heard of working forwards and
working backwards when solving problems, this is somewhat similar.)

§3.2 Working forwards

The philosophy here is pretty simple: come up with a problem, and try to solve it. Hopefully
you succeed, and the problem is interesting.
Unfortunately, it usually doesn’t work that cleanly. Sometimes, your problem will be intractible,
and other times it will be overly simple. When these happen, you’ll need to modify your setup
somehow to tune the difficulty of your problem.
As much as I don’t like to say it, it’s the ugly truth: being good at contest math helps. You’ll
have a wider class of problems that you can solve, and have more intuition for what kinds of
things are solvable.
There’s not much specific advice here, since this is simply a philosophy. Nevertheless, I’ll
provide a few of my own examples.

Example 3.1 (OMO Spring 2019/23)

Let a1 , a2 , a3 , a4 , and a5 be real numbers satisfying

a1 a2 + a2 a3 + a3 a4 + a4 a5 + a5 a1 = 20,
a1 a3 + a2 a4 + a3 a5 + a4 a1 + a5 a2 = 22.

Find the smallest possible value of a21 + a22 + a23 + a24 + a25 .

I got pretty lucky with this one: I had this problem idea at some point, and the problem was
solvable without any modifications.

Example 3.2 (USA TST 2019/6)

Let ABC be a triangle with incenter I, and let D be a point on line BC satisfying
∠AID = 90◦ . Let the excircle of triangle ABC opposite the vertex A be tangent to BC at
A1 . Define points B1 on CA and C1 on AB analogously, using the excircles opposite B and
C, respectively.
Prove that if quadrilateral AB1 A1 C1 is cyclic, then AD is tangent to the circumcircle of
4DB1 C1 .

This problem underwent some revision; I’ll outline the stages.

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Version 1. The point D was not present, and I simply asked to show that B1 , I, C1 are
collinear.
I thought this problem was cool, but that it hadn’t achieved its full potential: the problem
was a bit too simple. After some time, I came up with the following.

Version 2. The point D is now present, and the task is to show that B1 C1 bisects AD. I was
decently happy with this problem, until I noticed something which caused me to update the
problem again:

Version 3. The problem is updated to its final version (which you can see above, in the example
box). I updated the problem because I had observed that AD was tangent to the circumcircle of
4AB1 C1 ; combined with version 2, these combine to give the present problem.

§3.3 Working backwards

The philosophy here is basically the opposite: take an idea, and fashion a problem to that idea.
An example of this philosophy is the following:

Example 3.3 (RMM 2021/5)

Let n be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer
sides, perimeter 6n, and 60◦ rotational symmetry. In light of the pandemic, the government
of Zoomtopia would like to relocate its 3n2 + 3n + 1 citizens at 3n2 + 3n + 1 points in the
kingdom so that every two citizens have a distance of at least 1 for proper social distancing.
Prove that this is possible. (The kingdom is assumed to contain its boundary.)

This problem was constructed by taking an interesting idea, namely the fact that the kingdom
can be tiled with unit equilateral triangles and unit rhombi (with diagonals larger than 1), and
showing that this gives a valid tiling.
In this case, the idea could’ve stood by itself as a problem, but adding the extra layer on it
made it much better. (In some cases, the idea won’t be able to stand alone, and will need your
help to mold it into a problem.)

§3.4 Closing remarks

Constructing problems is hard, and you won’t know how effective you’ll be at it unless you try.
If I had to specify one thing which I think is a predictor of problem-writing success, I’d
probably go with whether you actively search for nice solutions to problems. In my opinion, if
you do so, then that means you’re focusing on the ideas more than just solving the problems in
front of you. That’s a good sign.
Of course, there are exceptions to this “rule.” I think it’s a big green flag, though.

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§4 General guidelines for problem creation

This section contains general guidelines for creating problems. Duh.
Important information: in general, problems have two components: the math portion, and
the presentation. (Some might call these the science and art components, but I prefer to not be
pretentious.)

§4.1 Make the problem look interesting

As such, there are two ways to make a problem stand out:

1. Make the mathematical content of the problem interesting.

2. Make the presentation of the problem interesting.

(This does not say “there are two ways to make a problem liked by others:” the easiest way to
do that is to ensure that your audience solves the problem, preferably after some struggle.)
The important idea here is that the second item is typically far easier to achieve than the
first. Achieving the first often requires a creative idea, which is typically not easy. On the other
hand, the second can require creativity as well, but typically an amount much less than required
for achieving the first.
As an example, consider the following two problems.

Example 4.1 (USEMO 2020/5)

The sides of a convex 200-gon A1 A2 . . . A200 are colored red and blue in an alternating
fashion. Suppose the extensions of the red sides determine a regular 100-gon, as do the
extensions of the blue sides.
Prove that the 50 diagonals A1 A101 , A3 A103 , . . . , A99 A199 are concurrent.

Example 4.2 (USEMO 2020/5, distilled)

Different circles ω1 and ω2 are fixed in the plane. Points P1 and P2 are chosen on ω1 and
ω2 , and revolve with the same angular velocity. Points Q1 and Q2 are chosen such that
P1 Q1 and P2 Q2 are diameters of ω1 and ω2 . Points P and Q are chosen such that P P1 and
QQ1 are tangent to ω1 , and P P2 and QQ2 are tangent to ω2 .
Prove that line P Q passes through a fixed point as P1 and P2 vary.

Which problem seems more interesting? While the second problem is perhaps more mathe-
matically solid, the first problem appears much more mysterious, and thus entices me to work
on it. (The first problem is also shorter, which is an important factor.)
An alternative approach is to create statements which seem simple, yet unexpected. While
the aforementioned USEMO 2020/5 fits the bill here as well, I think the following example is
more fitting.

Example 4.3 (USA TSTST 2019/9)

Let ABC be a triangle with incenter I. Points K and L are chosen on BC such that the
incircles of 4ABK and 4ABL are tangent at P , and the incircles of 4ACK and 4ACL
are tangent at Q. Prove that IP = IQ.

No talk about problem presentations would be complete without the following problem.

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Example 4.4 (IMO 2018/6)

Convex quadrilateral ABCD satisfies AB · CD = BC · DA. Point X is chosen inside ABCD
such that ∠XAB = ∠XCD and ∠XBC = ∠XDA. Prove that ∠AXB + ∠CXD = 180◦ .

§4.2 On answer extractions

Another important idea in problem design is that of answer extractions. Essentially, answer
extractions are the parts of the problem which allow the solver to show that they’ve solved the
problem.
This probably sounds like a bunch of hogwash, so I’ll offer some examples.

Example 4.5 (Shortlist 2019 N5)

Let a be a positive integer. We say that a positive integer b is a-good if an


b − 1 is divisible
by an + 1 for all positive integers n with an ≥ b.
Suppose b is a positive integer such that b is a-good, but b + 2 is not a-good. Prove that
b + 1 is prime.

The hypotheses and desired conclusion might seem quite arbitrary. In fact, the real meat of
the problem is the following claim.

Claim — The positive integer b is a-good if and only if

1. The integer b is even.

2. Every prime less than or equal to b also divides a.

Once this claim has been proven, the problem is fairly straightforward. Thus, the main difficulty
of this problem is conjecturing the claim, and proving it. If this problem had simply asked to
prove this claim, it would not be as interesting, since the main mystery of the problem is no
longer mysterious.
Problems can also have simpler answer extractions. A classical example is the following.

Example 4.6 (IMO 2013/6)

Let n ≥ 3 be an integer, and consider a circle with n + 1 equally spaced points marked on it.
Consider all labellings of these points with the numbers 0, 1, . . . , n such that each label
is used exactly once; two labelings differing only by a rotation are considered equivalent.
A labelling is called beautiful if, for any four labels a < b < c < d with a + d = b + c, the
chords ad and bc do not intersect.
Let M be the number of beautiful labelings, and let N be the number of ordered pairs
(x, y) of positive integers such that x + y ≤ n and gcd(x, y) = 1. Prove that

M = N + 1.

In fact, the desired statement M = N + 1 in this problem is also an answer extraction: the
real idea of the problem is to characterize the beautiful labelings; once the characterization is
found and proved, the N + 1 count is fairly simple.

§4.3 On contrived problems

My personal experience is that many people use “contrived” as a negative adjective to de-
scribe problems. To me, one of the most classical examples of a contrived problem is the

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following:

Example 4.7 (AIME 2014.I.14)

Find the largest real solution to the equation
3 5 17 19
+ + + = x2 − 11x − 4.
x − 3 x − 5 x − 17 x − 19

My opinion is that contrived problems are not intrinsically bad in any way. First of all, I’ll state
my interpretation of “contrived,” which might be different from other people’s interpretation.

Definition 4.8. A contrived problem is one where the problem setup is very specifically tuned
so that the solution works out. Taking the contrapositive, if we make small tweaks to the setup,
the intended solution will likely break.

Under this definition, most problems are contrived to some extent. For example, consider the
following AMC 8 problem, which received much praise for its creativity.

Example 4.9 (AMC 2015.8.25)

A plus-sign shaped polygon is obtained by cutting out 1 × 1 squares from the four corners of
a 5 × 5 square. What is the largest possible area of a square which can be inscribed inside
this polygon?

This problem is contrived, in the sense that the initial polygon (the plus-sign shaped one) is
crucial in the solution of the problem; with an arbitrary polygon, the problem would be far less
tractable.
The AIME problem above is very contrived, in the sense that changing any of the ten constants
in the problem breaks the intended solution. Nevertheless, I believe the AIME problem to be
decent; it tests for algebraic manipulation intuition and skill.
The following is an ultimate example of a contrived problem which enjoyed massive popular-
ity.

Example 4.10 (USEMO 2019/5)

Let P be a regular polygon, and let V be its set of vertices. Each point in V is colored red,
white, or blue. A subset of V is patriotic if it contains an equal number of points of each
color, and a side of P is dazzling if its endpoints are of different colors.
Suppose that V is patriotic and the number of dazzling edges of P is even. Prove that
there exists a line, not passing through any point in V, dividing V into two nonempty
patriotic subsets.

I call this is an “ultimate” example because the problem was entirely constructed backwards: I
had an idea for a solution, and constructed a problem which admitted this solution.
This problem could also be described as contrived in another sense: namely, the problem is
not phrased in its most natural state. When the problem is phrased in its most natural state, it
becomes quite straightforward. (The AIME problem above also shares this property.)

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Nevertheless, with all these things going against the problem, the contest community loved
the problem, and it became one of the most liked problems of the 2019-2020 cycle. So take this
as an example to show that problem creation isn’t entirely science.

Remark (Story on USEMO 2019/5). When I came up with the problem, I wasn’t expecting much,
and sent it to USEMO 2019, expecting the problem would be violently rejected for being contrived
at worst, and judged mediocre at best.
Imagine my surprise to see the problem on the exam, and my surprise at how many people liked it.
If I had known the problem would have been that popular, I would have sent it to the IMO instead.

§4.4 On computational problems

Many contests feature computational problems. For those who don’t know, a computational
problem is one which requires the solver to find an answer. Typically, this answer is a number;
sometimes, it can be more general, such as an algebraic expression. Often, the answer must be
an integer, for ease of automated grading.
The key distinction of computational problems is that they are typically graded based on
only the answer: all or nothing, depending on whether the answer is correct. This opens up
the possibility of a solver unrigorously arriving at an answer and getting the problem correct.
Typically, this is not desired, so answers for computational problems should not be
guessable.
In particular, many olympiad problems which require an answer have their difficulty in proving
said answer (not finding it). These problems do not make good computational problems. For
example, consider the following problem.

Example 4.11 (Erdös-Szekeres, special case)

Find the smallest integer n ≥ 10 such that for any permutation of 1, . . . , n, it is possible to
erase n − 10 of the numbers such that the remaining 10 numbers form a monotone sequence.

The answer to this problem is not terribly difficult to guess, though I wouldn’t call it easy by
any means. (It might be easier to guess if you think of the problem as constructing a long
permutation whose maximal monotone subsequences have length at most 9.)

§4.4.1 On computational answer extractions

If your computational contest requires specific answer formats (for example, AIME answers
must be integers between 0 and 999 inclusive), then you might face an issue: the answer to your
problem doesn’t fit that criteria. Is it not usable?
Not at all! You simply need to implement a way to get an answer of the desired format (an
integer, in this case) from the “actual” answer. The following might be one of the simplest
examples.

Example 4.12 (Answer extraction demo)

Let p be the probability that when a standard fair coin is flipped, it shows heads. Then p
can be expressed as ab , where a and b are relatively prime positive integers. Find a + b.

I like to use the term answer extraction for these mechanisms (e.g. find a + b, in the above
problem) as well.
There are many ways to implement an answer extraction, and there is typically no “right”
way to do it. As with the answer extractions discussed in 4.2, however, you should be thinking

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about your chosen extraction and how it could be improved. Figuring out how to do answer
extractions well is something you will gain from experience.
For examples of answer extractions, feel free to look at past AIME exams.

§4.4.2 Specific tips for computational problems

Here I give a few tips specific to computational problems. Duh.

Make the arithmetic simple Unless arithmetic is meant to be an idea in your problem, I
suggest simplifying the needed arithmetic for the solver as much as possible. It’s never fun to
get a problem wrong because of arithmetic errors.
To give a taste of how arithmetic could be an idea in a problem, I’ll give the following example,
which is hopefully self-explanatory.

Example 4.13 (Probably not original)

Compute
(22 + 122 + 222 + 322) + (678 + 778 + 878 + 978).

Safeguard against screw-ups This is a more advanced version of the above tip; for it to be
applicable, your answer format should be fairly restricted. (I’ll use the AIME answer format:
integers between 0 and 999 inclusive.)
The idea is to select the numbers in your problem such that the final answer “magically”
works out to your desired format. The hope is that, if the solver does the arithmetic incorrectly,
then they will get an answer not in the desired form, and so they automatically know they did
something wrong.
This is best illustrated via example, so let’s give a simple one:

Example 4.14 (Rhombus property)

Find the side length of a rhombus with diagonal lengths 16 and 30.

In this problem, the numbers 16 and 30 are carefully chosen so that the desired side length is
an integer. If a solver tries to solve the problem but miscomputes 162 = 236, then they would
obtain that the side length is
1 √ 1√ √
( 236 + 900) = 1136 = 284,
2 2
which cannot be expressed in the desired form, signalling to the solver that they did something
incorrectly.
Of course, you could combine this technique with traditional answer extractions.

§4.5 Never settle!

Often, you’ll have the urge to “finalize” your problem: you might decide that it’s not going to
get any better, and thus stop thinking about how to improve the problem.
This is a trap: you should always be working to improve your problems. As an example, I’ll
mention the first problem I ever got on a contest, from a time when I was far less experienced
with problem construction:

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Example 4.15 (USAMO 2018/4)

Let p be a prime, and let a1 , . . . , ap be integers. Prove that there exists an integer k such
that the numbers
a1 + k, a2 + 2k, . . . , ap + pk
leave at least 12 p remainders upon division by p.

This problem is okay in its current state, and would certainly be usable. However, thinking
some more about the situation in the problem, it is possible to see that the problem is false for
any odd composite p.
Thus, the following version of the problem could be considered.

Example 4.16 (What could’ve been)

Find all odd integers n ≥ 3 such that for any integers a1 , . . . , an , there exists an integer k
such that the numbers
a1 + k, a2 + 2k, . . . , an + nk
leave at least 21 n remainders upon division by n.

In my opinion, this is far superior to the original USAMO problem, and I wish I’d spent enough
time reviewing my problem to find it. Don’t be me.

§4.6 Some possible pitfalls

Here are some possible pitfalls on more specific types of problems. This section is mostly aligned
to my tastes and may offend some of you. You have been warned.

§4.6.1 On functional equations

A common technique for constructing functional equations seems to be the following: start with
an identity, and insert f into random places. Finally, try to solve the resulting equation.
In my experience, such problems are typically not interesting, and are best avoided. They
can be more interesting if they have unexpected solutions, but this is hard to ensure without
actually trying the functional equation. So, this is not an area I typically like to explore much.
Others may have more luck.

§4.6.2 On configuration geometry

Those who know me will know that I am. . . not the biggest fan of configuration geometry. I
think it can be done well, but many people constructing configuration geometry seem to be
fairly new to problem construction in general, most of these problems do not turn out well.
In particular, I think configuration geometry is done well when the final statement is short,
hiding the complexity of the problem. Three examples are the following:

Example 4.17 (Taiwan TST 2015, by Evan Chen)

In a scalene triangle ABC with incenter I, the incircle is tangent to CA and AB at E and
F . The tangents to the circumcircle of 4AEF at E and F meet at S. Lines EF and BC
intersect at T . Prove that the circle with diameter ST is orthogonal to the nine-point circle
of 4BIC.

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Example 4.18 (Shortlist 2019 G6)

Let I be the incenter of acute-angled triangle ABC. The incircle of 4ABC meets BC, CA,
and AB at D, E, and F . Line EF intersect the circumcircle of 4ABC at P and Q, such
that F lies between E and P . Prove that

∠DP A + ∠AQD = ∠QIP.

Example 4.19 (Shortlist 2020 G6, by Patrik Bak)

Let ABC be a triangle with incenter I and A-excenter IA . The incircle of 4ABC meets
BC at D. Define E = AD ∩ BIA and F = AD ∩ CIA . Show that the circumcircles of
4AID and 4IA EF are tangent to each other.

On the other hand, please don’t do the following:

Example 4.20 (Korea Finals 2019/2)

Let ABCD be a non-square rectangle. Point O lies inside 4BCD and satisfies OB = OD.
Points E 6= B and F = 6 D are chosen on lines AB and AD such that OE = OB and
OF = OD. Lines BF and DE meet at G.
Points X, Y , Z are the projections of G onto AB, BD, DA, and points L, M , N are the
projections of G onto CD, BD, BC. Finally, lines XY and M L meet at P , and lines Y Z
and M N meet at Q. Prove that BP and DQ are parallel.

§4.6.3 On olympiad-style computational inequalities

This might be obvious, but I’ve seen it just enough that I’d like to specifically mention it.
Most olympiad inequalities involve equality cases when the variables are all equal. This make
the problem very guessable, so these problems are typically not great.
Some inequalities have equality cases that are less guessable; these usually turn out better.
An example would be the following:

Example 4.21 (IMO 2006/3)

Determine the least real number M such that

|ab(a2 − b2 ) + bc(b2 − c2 ) + ca(c2 − a2 )| ≤ M (a2 + b2 + c2 )

holds for all real numbers a, b and c.

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§5 General guidelines for contest creation

This section deals with preparing problems for contests.

§5.1 Format your problems nicely

This is the main one. Make sure your contest documents are as well-formatted as possible. When
someone opens your contest document for the first time, the first thing that will stand out is the
formatting – if the formatting is unappealing, then one might be less inclined to take the contest,
as they might believe that the organizers did not put enough effort into running their contest.
Looking at it more positively, seeing a well-formatted contest, the contestant might recognize
the effort put into the contest, and be more inclined to participate in it.

§5.2 Formatting tips

Here is a collection of formatting tips.

§5.2.1 On math formatting

There are a few things to cover here.

Ambiguous function powers Consider the notation f 2 (x). Some use it to denote f (x)2 , while
others use it to denote f (f (x)).
My preference is to always use f (x)2 for the square of f (x), and to always use f 2 (x) for
f (f (x)); this way, both forms of function powers are representable. I suggest you use it as well.
When administering a contest to any reasonably sized group of people, if the f k (x) notation
is used, it should always be explained, since in my experience this custom is not universal.

Natural numbers Many countries use N to denote the integers at least 1, and many countries
use N to denote the integers at least 0. Some countries in the former category use a notation
such as N0 to denote the nonnegative integers.
Thus, I suggest to simply not use N at all; not even with a definition such as the following:

Here, N = {1, 2, . . . } is the set of positive integers.

There are a few alternatives.

• Use the notations Z>0 and Z≥0 . These are unambiguous, and so carry a smaller risk of
confusion. (However, these notations are not standard, and so you should define them if
they are used.)
This is the custom typically followed by the IMO and IMO shortlist, and I support it. The
main complaint against them might be that these symbols are much more similar to each
other than N and N0 are, which increases the risk of people misreading the problem.

• Simply write out the sets, as say {1, 2, . . . } and {0, 1, 2, . . . }. This method is usually
cleaner than the above method, but is not entirely rigorous: the definitions of the sets
might not be entirely clear. As such, I do not usually use this method, and especially not
in high-stakes contests or contests where contestants cannot ask for clarifications.

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§5.2.2 On typesetting
Use good LaTeX style. This is quite general, so I’ll give a few sub-tips corresponding to the
typesetting oddities I see the most. Of course, these are not meant to be exhaustive in any way.
Most of these are quite pedantic, but they’ve become things I notice often, so I’ll mention them.
On the other hand, I am not a typesetting expert, and so would appreciate any feedback on
good style.

• Use \cdots and \ldots instead of ..., as well as other ellipsis commands. In most
situations, one of these looks much better than the other. For example,

{1, . . . , n} and 1 + ··· + n

look much better than

{1, · · · , n} and 1 + . . . + n.
In many situations, \dots will work and adapt to the specific situation it’s used in.

• In problems involving functions (often functional equations), to introduce a function with

its domain and codomain, use \colon instead of : (a colon character).
The former results in f : Z → Z, and the latter results in f : Z → Z. (Note the spacing
between the f and the colon.)

§5.2.3 On English
Use good English, including correct grammar. (Of course, if you’re developing a contest in
another language, the same applies for that language.)

§5.3 On official solutions

It goes without saying, but the official solutions should be as polished as the problems. In
particular, you should have the solutions well-developed before the contest even starts. (This
also helps catch any errors in the problems.)
In fact, my preferred guideline is that a problem is not even eligible to be used unless it has
a full solution submitted by the author. It removes a lot of the burden from you, the contest
organizer.

§5.4 On releasing problems not used for contest

Many contests release problems which are not part of the actual contest. These might be “sample
problems” released before the contest in order to prove to the masses that the contest is worth
looking at. Another very common occurrence is for contest organizers to release shortlists of
unused problems.
I think these customs are terrible, and dislike that they are typically expected of contest
organizers. All these customs do is remove perfectly good problems from considerations from
future contests. In addition, these problems do not typically receive much attention, leading to a
sad death for them. I’m eternally grateful USAMO and USA TST(ST) do not release shortlists.
Many proposers do not want their problems to suffer this fate; I am certainly one of them. A
common issue that arises is that problems become public (through the shortlist being released)
while the proposer is unaware. Take a look at how many RMM extras problems were used in
other contests because the proposers assumed they were confidential.
Thus, if you must release a shortlist, you must ask permission from the problem proposers to
include their problems in a shortlist. Not doing so is quite disrespectful.

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§6 Hall of fame
This section lists a few problems that I think are exceptional from a design standpoint. They
might not be the most interesting problems mathematically, but they are problems that I wish I
had come up with.
This section is fairly small currently, but will hopefully be expanded as I come across more
problems.

Example 6.1 (Unknown source)

Let p be a prime and let a, b, and c be positive integers satisfying a < b < c < p. Suppose
that a3 , b3 , and c3 are all congruent modulo p. Prove that a + b + c divides a2 + b2 + c2 .

I really like the ending of this problem: it throws an unexpected catch at you, and the resolution
is quite nice.

Example 6.2 (NIMO April Fools 2013.8, Lewis Chen)

A person flips 2010 coins at a time. He gains one cent every time he flips a prime number
of heads, but must stop once he flips a non-prime number. If his expected amount of
money gained in dollars is ab , where a and b are relatively prime positive integers, compute
dlog2 (100a + b)e.

I think the above problem is one of the best computational problems ever made. Saying anything
more would spoil the problem; I highly suggest you try it for yourself.

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