This post is based on a talk I gave at AwesomeMath Cornell 2017 Session 1 (though not Session 2, since the program coordinators wanted to make room for presenters who did not speak during Cornell 1). Due to growing much longer in length than anticipated, this has been split up into multiple parts. This is the first part.

One of the main reasons why contest mathematics is so popular among middle and high school students is because it allows for them to solve really interesting problems they would not be able to approach otherwise. Ideally, a math competition question should challenge but not daunt. It should highlight how crazy and innovative mathematics can be when stretched to its limits, and it should invoke a feeling of happiness when solved (either in practice or during the actual contest). The problems in the American Mathematics Competition series of contests (AMC 8/10/12, AIME, USAMO) have long been ambassadors for this mentality – this is after all one of the reasons their popularity is so massive at the time of writing.

But writing problems like these is somewhat of an art form. Just as how one needs practice to get good at problem-solving, one needs also persistence and patience to write problems which are of equal quality. Although writing such problems is very difficult (even for veterans of the craft), it is not impossible, and at the same time a very rewarding experience.

With that said, this post gives some of my experiences with writing problems for various contests. My aim is to discuss various tips and techniques one uses to write problems, based off of both my experience and of examples from others. I will also give examples of these techniques being used in practice along the way.

As a warning, this post contains spoilers for a few problems from past NIMO and CMIMC contests.

1. Why Write Problems?

Anyone who browses the Art of Problem Solving fora at any point during the year knows that mock (i.e. practice) contests are very prevalent among the contest community. But it is not immediately clear that doing so gives benefits – writing problems takes away time one could be using to instead solve problems and get better at mathematics in general. Why, then, do so many people write problems?

Many people say that writing problems can also help make people better at mathematics, because in some sense writing problems is much harder than simply solving them. While this may be true in the strictest sense of the word, in practice I disagree with this. While writing problems can be helpful, most of the time this is considered playing around, and is usually not what one thinks of when writing for competitions. When writing for contests, one usually constructs problems at a level which he or she is comfortable already, and so from this perspective writing questions gives a questionable return on investment. (In other words, one is not going to write a practice AIME contest solo if said person is still learning how to solve AIME questions.)

That being said, every person has his or her own motivations for writing problems; here are some of mine.

The ideas you work on are more interesting. When solving problems on competitions, you have no control over the quality of the problem presented to you – one of the more annoying aspects of math competitions is being forced to solve a problem you have no interest in. When writing problems, in contrast, you get to work on ideas which you find appealing. I think Alexander Katz words this pretty well: “It’s just way more interesting to be writing problems than solving them: when I’m solving a problem, I don’t really know if it’s an interesting problem or not until I’ve essentially found the solution, which leads to lot of ‘wow this is sick’ moments but also a lot of ‘that problem sucked’ moments. When writing problems, almost everything is interesting by default, since uninteresting things get tossed out the window pretty quickly. The thrill of discovering a problem and its solution is at least an order of magnitude greater than just discovering a solution.”

When solving problems on competitions, you have no control over the quality of the problem presented to you – one of the more annoying aspects of math competitions is being forced to solve a problem you have no interest in. When writing problems, in contrast, you get to work on ideas which you find appealing. I think Alexander Katz words this pretty well: “It’s just way more interesting to be writing problems than solving them: when I’m solving a problem, I don’t really know if it’s an interesting problem or not until I’ve essentially found the solution, which leads to lot of ‘wow this is sick’ moments but also a lot of ‘that problem sucked’ moments. When writing problems, almost everything is interesting by default, since uninteresting things get tossed out the window pretty quickly. The thrill of discovering a problem and its solution is at least an order of magnitude greater than just discovering a solution.” The time constraint is removed. One of the main complaints about contest mathematics is the speed component. Most contests require students to solve challenging problems in a very small time frame (with USAMTS being one of the lone exceptions). When writing problems, however, you have the freedom of expanding on ideas for longer periods of time. This allows you to construct more intricate problems and play around with more complicated ideas, which in turn can help you refine them until you get a working problem idea. Problem writer Evan Chen has reiterated this idea in the past: “When you’re writing problems, you actually get to choose what you work on, you don’t have a time limit, and you can use any aids you like. For example most olympiad geometry problems I propose are things that I come up with after 3-6 hours of playing around with GeoGebra, and are not things I could come up with any reasonable contest environment. It’s nice to have some of the artificial constraints lifted.”

One of the main complaints about contest mathematics is the speed component. Most contests require students to solve challenging problems in a very small time frame (with USAMTS being one of the lone exceptions). When writing problems, however, you have the freedom of expanding on ideas for longer periods of time. This allows you to construct more intricate problems and play around with more complicated ideas, which in turn can help you refine them until you get a working problem idea. Problem writer Evan Chen has reiterated this idea in the past: “When you’re writing problems, you actually get to choose what you work on, you don’t have a time limit, and you can use any aids you like. For example most olympiad geometry problems I propose are things that I come up with after 3-6 hours of playing around with GeoGebra, and are not things I could come up with any reasonable contest environment. It’s nice to have some of the artificial constraints lifted.” Nice ideas are meant to be shared! As Michael Tang notes, “[I]t’s fulfilling to make ideas come to life and create a finished product that people can enjoy; besides, it’s an opportunity to give back to the math contest community.” Coming up with an interesting idea that works well is a very rewarding experience, and many experienced problem writers are proud of their finest creations.

But perhaps the above has a second, more hidden reason behind why some people (including me) write problems. As I stated earlier, problems on math contests (e.g. AMC/AIME/USAMO, ARML, and college contests) are meant to invoke enthusiasm in problem solving and reward contestants with the experience of solving difficult and creative problems. But it is also important to keep in mind that these problems are written by people as well. As such, eventually old problem writers must cycle out and be replaced by new ones. It is important, then, that the spirit of problem writing is kept alive so that future generations of math contestants are able to have these same mind-blowing experiences. (This is one of the main reasons why I personally advocate for a smoother transition between being an AMC/AIME/USAMO contestant and writing problems for said contests – newer minds often have fresh ideas and experiences that can translate very well to the problem writing side of the spectrum.)

2. An Origin Story

Before we dive into the details of how to write problems, I think it might be instructive to explain my problem writing history. I think it touches upon a few important ideas that are hard to explain otherwise.

It is hard to pinpoint exactly where my problem writing ”career” began – I was known for being a somewhat inquisitive kid back in elementary school, before I had any tangible idea of what a math contest was. However, I personally attribute it to 5th grade, when I was first exposed to a computer game called Descartes Cove, produced by/affiliated with Johns Hopkins CTY.

The basic idea behind this game is simple. You, an unnamed adventurer, find yourself stranded on an island, and upon arrival only find a chest with a few items of interest (a backpack, a map, etc.). Can you explore the island, solve math problems, and collect enough parts to build a means of escape? The game is split into six disks, each exploring a different part of the island and in turn emphasizing a different aspect of math (in order: Measurement, Number & Operations, Data Analysis & Probability, Algebra, Geometry, and Reasoning & Proof). Although this may seem cheesy, the presentation made the experience very engaging; to a fifth grader, this was all I needed to be hooked. In fact, I liked the game so much, that I said to myself ”I want to make a Descartes Cove 2”!

Of course, not knowing anything about game design, the only thing I could do was write problems. But unfortunately, I could not do that either. My easy problems were probably somewhat okay: standard algebra questions (e.g. ”how many books could you fit on a bookshelf?”) that one could realistically see in an early AMC8. My hard problems, however, were exactly the same as the easy problems, except that I made the scenarios arbitrarily complicated and converted all the numbers to nasty decimals. Oops.

Although the problem quality at the time was extremely sub par, it was a start, and thus began the journey of writing problems for enjoyment.

Through the next five years, I began taking problem writing more seriously, starting from simple practice contests and moving up all the way to real ones. Here is a list of personal ventures in chronological order.

Mock Number Theory MATHCOUNTS Sprint Round (solo). This I do not consider to be an official project of mine, but I think it is interesting to see where I was back in 8th grade. It is worth noting that most of the problems here are either classical or slight modifications of Alcumus problems (e.g. #30). It is also worth noting that at the time I found problems such as #29 quite hard.

2012 Mock AIME II (me, Jaenic Lee, Kyle Gettig, Justin Stevens, Alex Katz, Siddharth Prasad). This was my first major mock contest, and as such I was somewhat inexperienced in problem placement; I particularly remember trying to submit the current #1 as a #15 and subsequently watching it slowly move down over the course of a few hours.

2012 Mock AMC 12 (same as above). I happened to write a ton of the questions here, but almost all of them were in the first half of the test (and those that were not might have been a bit too high on the test).

2013 Mock AIME I (me, Kyle Gettig, Justin Stevens, Siddharth Prasad, Kevin Sun). Only two of my proposals were accepted, but notice that my problems are starting to get a bit harder.

2013 Mock AMC 10 and 2013 Mock AMC 12 (solo). This duo of mock tests was my first and only venture in which I wrote the entire test by myself. I personally think this is the contest in which I started developing an identity as a problem writer; although it was not perfect, it was very well received by the people who took it.

2015 Mock AIME I (me, Justin Stevens). This was the last mock contest Justin and I helped write, and we both consider it to be our best by far. (An interesting fact about this test is that it was originally designed as a Mock Mandelbrot competition, since this year would be the first year that the test entered its three year hiatus. Speaking of which, it will finally be back and running this year, and I highly encourage schools to register for it!)

National Internet Math Olympiad (me, Evan Chen, Justin Stevens, Michael Ren, Michael Tang, etc.; 2014-“present”). This perhaps marks a major transition in my problem writing career, in which I was able to write problems for the first time for a semi-serious contest. My senior year in high school was when I dedicated the most time to writing for this contest, since at the time it was my primary problem dump; I happen to think that having this as a breeding ground was very helpful, since it gave me an ultimate purpose to continue proposing problems and get feedback on them throughout the year. Thanks Evan for putting your trust in me!

CMIMC (2016 – present). This was a contest a few friends and I started before/during our freshman year at Carnegie Mellon University (although efforts had been made in the past), partly in response to CMU’s growing reputation as a mathematics school. Fortunately, it is still going strong!

AMC10/AMC12/AIME (2016-present). Pretty self-explanatory. Although I do not have any problems on here yet, my first batch will be arriving in 2018 – I hope test takers find just as much enjoyment in them as I do!

There are a few things to note about the above list.

Almost all my projects involve multiple problem writers. This is very much intentional: it is much easier to write a few good problems for a contest than it is to write an entire contest spanning various subjects and difficulties. This was especially important in my very first ventures: at the time my mathematical ability was nowhere near the level it would need to be to write a mock AMC or AIME myself, so instead I saw these mocks as places to submit a few problems I had written in order to test the waters a bit.

The projects I worked on grew more and more serious, but at a very gradual pace. The time span between my first mock contest and my first AMC submissions was five years. It took a lot of patience and practice writing questions to get to the point where I could say my problems were legitimately good. This adds strength to the “patience and persistence” clause mentioned above: if I had no experience, I would not have the confidence necessary to propose good problems to contests nowadays.

3. Problem Writing General Techniques

Now that we have established a bit of background, let us dive into some techniques on how to write problems. There are generally two broad categories people tend to bucket problem writing in: forwards and backwards. We now explore both of these.

3.1. Writing Backwards

This is the first general method by which people write problems, and is perhaps the more common one. The basic idea behind writing problems backwards is simple: you come up with an idea for a problem and then back-construct the problem around this idea. For example, one may discover a clever application of the Cauchy-Schwarz inequality and then write a problem whose official (and perhaps simplest) solution revolves around noticing this application.

There are a few pros and cons with this method of writing problems.

PRO: You have control over the solution to the problem. Since you are the one designing the problem in this method, you know exactly how the intended solution should look. For example, as discussed above, the solution to the Cauchy-Schwarz problem (if said problem is constructed with enough care) is nice enough that people may think the problem is interesting and instructive.

Since you are the one designing the problem in this method, you know exactly how the intended solution should look. For example, as discussed above, the solution to the Cauchy-Schwarz problem (if said problem is constructed with enough care) is nice enough that people may think the problem is interesting and instructive. PRO: It is (relatively) easy to construct problems this way. Explanation of this bullet point will be deferred to the next subsection, but it is still worth mentioning now.

Explanation of this bullet point will be deferred to the next subsection, but it is still worth mentioning now. CON: It is harder to gauge problem difficulty. Remember, you are the one who has come up with the trick independent of seeing the problem. Thus, it is probably very likely that you think the problem is easier than it actually is, since to fresh eyes the trick may be unmotivated.

Remember, you are the one who has come up with the trick independent of seeing the problem. Thus, it is probably very likely that you think the problem is easier than it actually is, since to fresh eyes the trick may be unmotivated. CON: These types of problems often seem artificial. Basing the problem around a certain technique may impose weird constraints on the variables in a problem or may otherwise impose an unnatural setup that might not seem appealing. I am not suggesting that all problems are like this – indeed, it is possible to construct problems backwards that do seem very natural. However, I would advise to be careful!

Unfortunately, most of my problems are not written backwards – I have a stubborn tendency to write problems in a more forward direction. Thus, I will be content with giving only one of my own examples as an illustration of this technique.

Example 1 (NIMO 15.6) For all positive integers , define . Compute the largest positive integer such that

As one may notice when solving the problem, the key to this question is the factorization

This is intentional! My goal in writing this problem was to construct a question revolving around this interesting identity.

After playing around with the resulting setup (perhaps influenced by previous problems I had done), I noticed that this identity could be rewritten as

which, upon setting , rewrites as

This was nice enough that I then immediately decided I wanted the problem to involve telescoping in some way. My first choice was telescoping sums, but I could not get the mathematics to work out in a way which I found pleasing. However, telescoping products did end up working, and the result (after introducing a bit of symmetry) led to the problem you see above.

Before we move on to writing problems forwards, it is worth addressing the last bullet point more closely. Consider the following two problems from recent AIME examinations:

Example 2 (AIME 2014.I.14) Let be the largest real solution to the equation There are positive integers such that . Find .

Example 3 (AIME 2016.I.7) For integers and consider the complex number Find the number of ordered pairs of integers such that this complex number is a real number.

I would venture to guess that both of these problems were written in a backwards manner and in some sense seem contrived. But the reaction surrounding these problems was very different! Many contestants (including myself, who solved it live) saw the first problem as a very creative algebra question, requiring a lot of nice intuition in order to solve it. On the contrary, the second question was almost universally hated. I will defend the problem writer here a little bit: I think the idea behind this second question – namely that just because a complex number is in the form does not imply that the real part is and the imaginary part is – is not completely terrible. However, the problem was not constructed with enough care, leading to a problem which (a) felt completely unnatural and (b) was somewhat controversial (since many contestants thought that was necessarily positive, when in reality the official solution treated as for ). This reiterates what I mean about being careful when constructing a problem backwards.

3.2. Writing Forwards

This problem writing technique is somewhat different in nature. Instead of crafting a problem around an intended solution, one starts with an interesting idea and sees where it takes him or her. In effect, this is more akin to mini mathematical research, because you are chipping away at unknown territory and trying to see whether any interesting ideas pop out.

The positives and negatives of writing problems forwards mirror those for writing problems backwards quite well.

PRO: Problem statements are often more natural. Pretty self-explanatory. By focusing on an interesting/natural idea and seeing where it takes you, it is more likely that the resulting problem you come up with is also natural.

Pretty self-explanatory. By focusing on an interesting/natural idea and seeing where it takes you, it is more likely that the resulting problem you come up with is also natural. CON: It is much harder to write problems this way. In many cases these types of problems are restricted by how interesting the results turn out to be – which is something you do not know ahead of time. Thus, it is often the case that the probing yields unsatisfactory results, meaning that you will have to shut the idea down (at least for now). At the same time, though, when you do find something which words, the experience is highly rewarding!

In many cases these types of problems are restricted by how interesting the results turn out to be – which is something you do not know ahead of time. Thus, it is often the case that the probing yields unsatisfactory results, meaning that you will have to shut the idea down (at least for now). At the same time, though, when you do find something which words, the experience is highly rewarding! CON: You have less control over the solution to the problem. Again, pretty self-explanatory, and this ties in pretty well with the above bullet point.

I think an instructive example of this comes from the hardest problem on the 2016 CMIMC Geometry test.

Example 4 (CMIMC 2016.G.10) Let be a triangle with circumcircle and let be the midpoint of the major arc . The incircle of is tangent to and at points and respectively. Suppose point is placed on the same side of as such that . Let intersect at a point . Given that , , and , compute .

About a month before the contest, my goal was to write a problem that was harder than #9 (which was at the time the presumptive final problem on the geo test). I started off with a triangle and its circumcenter . Inspired somewhat by 2008 USAMO #2, I drew the circumcircle of and extended to meet at . Looking at this diagram for a while, I eventually noticed that was the midpoint of , where is the antipode of with respect to , and that . This in turn meant that was in fact the incenter of (and the -excenter). Then looking at as the reference triangle, I had the following restatement of a key step in the USAMO problem:

LEMMA: Suppose that is the -excenter and the midpoint of major arc of a triangle . Then is the -symmedian of .

Unfortunately, I then realized that this property was very easy to prove by just considering the excentral triangle of (aka the Big Picture configuration in 107). Welp. I tried to find a way to hide this further, but I didn’t get anything satisfactory, so I put the problem down.

Fast forward two weeks. I start fresh, looking at the incircle of a . I am reminded from various other problems that if is the point of contact of the incircle with , then is a symmedian of . Strange – we have a symmedian, and since (where and are the other two tangency points), can be considered the midpoint of the arc of some circle.

Then the idea came: what if I found a triangle with circumcircle for which was the -excenter?

By Fact 5, it suffices to extend and intersect it with at ; then is the desired triangle. I play around with some angle chasing and find that triangles and are similar! At this point, I realize I have an interesting configuration to work with.

I then spend the next several hours on Geogebra, constructing the diagram and seeing what I could find. I noticed two key observations during this period. After that, I tried to see if I could ask something which required both observations. A bit more experimentation yielded what you now see as the G10.

Then contest day came, and nobody solved either #9 or #10. Oops. At least I got a nice-looking diagram out of it.