Escape Velocity

The current state of mathematics competitions is incredibly unwelcoming to beginners, especially high schoolers. Here are my thoughts on why this is the case, and how to alleviate this problem.

Starting Costs

It takes an incredible amount of energy for high school students to actually get into the competitive mathematics scene. This problem doesn’t exist for middle schoolers for a couple of reasons.

  • The competitions are more simple so theory is minimal. The problems, though usually boring, are approachable.

  • The lack of a good way to train for MATHCOUNTS makes it so that the only way to practice is to do problems.

  • There’s usually an actively involved coach and a larger peer group (since entry costs are low).

But when you’re in high school none of these are true anymore. Local competitions are not a reliable introduction to competition math, so the AMC is the only introduction to competition math that’s cheap, convenient, and reliable. But the AMC is a very harsh introduction to competition math, especially for juniors and seniors. The introductory questions are either too boring, too weird, or too hard. 2019 was a particularly egregious year – here you have beginners hoping that maybe they’ll be able to do a couple of questions and get around 80 points even though pretty much two thirds of the test is inaccessible to them. Then two weeks later score reports come back and they learn that they scored 60-70 because of stuff like this.

This is very discouraging and can very quickly kill interest. Although in the long run almost everyone who became notable enough to speak says that learning how amazing their competition was good for them, I suspect survivorship bias is at play here. There are a lot of times when this kind of stuff gives off the undesirable message “you suck” and completely kills off a kid’s interest. This actually happens surprisingly often, and the most common way to counteract this is to be dragged through it by force until you’re past escape velocity. Often this role is played by the parents and it ends up being a balancing act where the parents have to make their kids try something, but if they push too hard any interest is completely killed.

It turns out that parents are actually really bad at fulfilling this role. But it helps tremendously to have a friend dragging you along to do something that matters. (Note that things that matter don’t have to be “important.” Even if admission officers and your parents don’t care, I’d argue that your friends asking you to hang out/bake cookies/water the school garden/etc are much more real and matter more than stuff like homework or filling your life with empty extracurricular activities. I wonder what would happen if the best way to get into college was to just live your life fully.) If you hang out with smart people and want to do a smart person thing, chances are one of your friends are involved in it. So force them to take you along for a ride. It’ll be helpful for them too. And if your friend insists on dragging you along – seize the opportunity. This will help you significantly lower the starting costs of getting into something meaningful. (This is one of the many reasons smart people like to hang around other smart people.)

One of the biggest issues is that few people have externalized the way they learn math. And very few of them know how to recreate the experience of getting into math competitions. It’s very easy for math competition kids to get into physics, chemistry, computer science, and so on is because USACO has a very smooth learning curve by nature, and physics/chemistry competitions have their material covered at school. But it’s very hard for those physics kids to get into math competitions because there is no good introduction to math competitions. The introduction is thrilling enough

I think the only long-term solution to this problem is to create a low-stakes introduction for high schoolers that’s thrilling enough to get beginners to come to the scene, similar to MATHCOUNTS. A previous version of the draft had “that feels important enough,” but I think this is a misleading way to think about it. I think deep down middle schoolers know that they’ll have to start contributing to the community in some way eventually. The realization hits them just late enough so it doesn’t scare them away from the scene, and I think this is the best of all worlds. And the thing about MATHCOUNTS is it’s also cheap (on your side) and convenient – you make the tryouts for your club (if there are any to begin with) and your coach will take care of all the logistics for you.

The biggest problem is for those without a community of other smart kids in their geographical area. For these kids all of these problems have always been there, and as an added bonus, they get to deal with them without guidance. For these kids I highly recommend going to a summer program – you’ll experience what it’s like to have a group of really smart kids and you’ll be able to stay connected to them to some degree. But I live in a big competition math area, so I can’t really help much in this regard.

The reason math competition isn’t really able to spread out of its hearths is because it’s really easy to lose people along the way. We need to do a better job of reaching out to people and helping them find their way.

Introductory Texts

I’ve tried to get into several competitive academics, including debate, physics, chemistry, and computer science (basically the entire list). I think maybe one of the reasons I had trouble doing this is because the introductions are very abstract and philosophical. For example, Learn C++ takes this up to eleven. (I actually started getting the motivation to do this is Perhaps something similar is happening with math for a lot of kids; they’re having trouble getting into it because there’s no action. There’s no excitement. It’s just a bunch of explanations about a topic, and I think increasingly often kids just get bored and stop reading at some point, and they have to actually force themselves to continue reading until it gets interesting.

In fact I think this is a large reason there’s such a huge gap in mathematics competitions; most people who’ve made AIME can honestly say “I felt like it was just going to happen,” while many people just fall behind and lose interest. It isn’t a gap in intelligence – it’s a gap in interest. The people who’re drawn to math by some combination of curiosity and enjoyment are far likely to do better than the people that are doing it because they feel like they should for a useful reason. The qualifier “for a useful reason” is very important – speaking personally, there are times when I haven’t worked on math for a while and feel like I should do math because not having done it for a while feels wrong. Not wrong because “I’m not spending my time productively,” because I get this same feeling when I work on homework for too long, much to the chagrin of my grades. Wrong because I haven’t done math. (I think this is maybe why a lot of people also don’t breach the computational to olympiad barrier, because they perceive it as separate subjects and philosophy is really prevalent during olympiad.) The hardest part about getting into a competitive academic is usually having enough interest to force yourself through the beginning – the momentum will carry you from there.

But here’s the issue with just “getting rid of the fluff” – it isn’t fluff. This type of philosophical stuff is actually important. It’s also general, which is why the best place for it to be is in the front. In fact the same issue exists with the theory for a handout, and this definitely isn’t “fluff.” So what can textbook designers do, and what can students do?

For textbook designers, make the philosophy and theory feel out of the way. The problems are what matters. Theory is just incidentally a toolbox you can refer to, and philosophy is just a rough idea of “how and when should I use these tools?” Make your design reflect this. Publishing companies: take care to physically design these things to be out of the way – they can be referenced at will but do not take the focus away from the problems. Student teachers: Make your theory feel out of the way by making it as concise as possible.

And make your text naked. What that means is you don’t need to add filler sentences or whatever to inflate page count, or to make the text not feel naked. If what you’re writing is worth reading, there’s no need to dress it up. In fact it’s much easier to understand when it’s not dressed up. Present your conclusions the same way you thought of them, but streamlined. In this way what you write will reflect reality. As an example, consider Projective Geometry in math olympiads. Every text will explain the projective plane, cross ratios, and harmonic bundles. Then you get what feels like a “configuration dump,” where you have a bunch of configurations that imply harmonic bundles. This is the part where inexperienced writers would try to dress it up. (This doesn’t happen because in this specific case because inexperienced writers will generally not write about projective geometry.) But in fact, laying it bare as a configuration dump is not only more concise, it signals to the reader that it is a configuration dump. When you’re writing, just make sure what you’re writing about is significant – being interesting will naturally come with being useful.

In particular, put “skip” tags on skippable stuff. (This includes useful stuff that can be returned to later when the interest barrier is cleared.) If there’s something you honestly think someone who knows it already can skip, communicate this. Either do this implicitly by putting this in an “Extra” section, or a Chapter 0

For students, I recommend the following four step process.

  • Skim the theory so you have some minimal idea what’s going on.

  • Do problems (and inevitably fail).

  • Go back and read the theory while looking for something that solves the problem(s) that you failed on.

  • Repeat until the theory becomes second place to you.

I must emphasize that this will not work unless the problems are significant. So if you’re reading a textbook filled with boring and routine problems, you’re reading a bad textbook and you need to put it down. A good first order approximation of interesting problems is if they’re from a reputable competition, and a good second order approximation of interesting problems is if they feel like they could be from a reputable competition.

Flexibility

One of the most important life skills is to be able to adjust and adapt any aspects of your plans when they are failing to achieve a goal. The rigidity of school prevents this.

This is an adaptation of a speech I wrote for TedX. It has been changed to fit a more mathematically inclined audience.

Broken Toilet

Let’s say one day you tried to flush a toilet and nothing happened. Suspecting that something might be wrong, you press the lever again and nothing happens. You press the lever a couple more times and still it refuses to flush. At this point a reasonable person would conclude, “The toilet is not working” and call over a plumber or fix it themselves if they know how.

Now consider a student who cannot solve probability problems about the Binomial Theorem. Consider some random school textbook that has the following explanation

• Binary? The possible outcomes of each trial can be classified as
“success” or “failure.”
• Independent? Trials must be independent; that is, knowing the result
of one trial must not have any effect on the result of any other trial.
• Number? The number of trials n of the chance process must be fixed
in advance.
• Success? On each trial, the probability p of success must be the
same.

and presents problems like these ad nauseum.

A coin is tossed 10 times.  What is the probability that it shows exactly 3 heads?

Boring. I would only present something like this twice (once for the worked example and another for the check-up). At this point we get it, and want to actually do something interesting with a concept for once.

The materials are scattered, the textbook presents a total of zero worked examples, and the student doesn’t know where to actually find good materials. He tries to read the textbook but can’t understand it, and tries to do some problems but doesn’t know how. So what does the average student conclude? Instead of concluding that “This textbook is not working and my teacher’s lecture makes no sense, so let’s fix the problem of having bad material,” they instead conclude, “This is too hard for me. I’m not good at math. I’m just not one of the smart ones.” And society completely reinforces this message.

(Side note: I had no idea this was how school taught Binomial Theorem until now. I was an ardent believer of having schools actually teach combinatorics, especially as it is the topic with the least theory, but seeing what it has done to middle school competition math has changed my mind. I’m not saying that having clear criteria is always bad. But if you’re introducing people with the mathematical maturity of a 6th or 7th grader in the Math Club to counting, you’re much better off saying “this is what counting problems feel like, go do some,” especially when counting is the subject about intuition.)

Slow Change

Let’s say you tell a technician your computer isn’t working because you got a virus installed on your computer. They would probably get it fixed in a timescale of days. Now let’s say you tell your teacher that the textbook isn’t working for you. It’s boring, stupid, whatever. It would probably get fixed on a timescale of never.

In fact, even if there are many legitimate problems, there will not be a substantial change. “But Common Core works fine for us” is not a valid defense either – nobody should have the power to decide how other people teach when they are so far removed. (In fact, I think it is explicitly harmful for people who can prompt change through an uninformed opinion to be expressing them. This is probably one of the more altruistic reasons celebrities do not make a big deal of political opinions.)

But let’s say customer support sucks and your computer doesn’t get fixed. What then? The wrong thing to do is to throw a fit at the world and complain about how unfair it is or how stupid it is that these people cannot do their job. You’ve got to fix the computer by yourself. It is absolutely unfair that the state of education is what it is and it is worth fighting to fix it. But you’ve got to fix the computer in the first place, and there are plenty of nice people willing to help you do this.

The smart ones

I think a big misconception people have about “smart kids” is that the reason they can skip so far ahead in school math or take 6 APs without caring or whatever is because they know more stuff. It’s because they’re used to a specific set of well-defined tasks. (It does help that much of the stuff taught is repeated in 3 or 4 different classes, but I maintain that this is not the primary reason.)

The difference is more like asking someone who has never really fixed anything to fix your toaster versus someone who works in an auto repair shop. Even though neither of them has fixed a toaster, one of them will be done much before the either.

(This is not implying that getting good grades becomes useful in life later. I just couldn’t think of anything genuinely useless for this analogy. See Paul Graham for why learning to get good grades is actively harmful.)

Shelf it for later

But even the broken toilet cannot fully capture the sheer absurdity of this situation. Unlike a clogged toilet, which may present some problems unless dealt with in a timely manner, not learning a certain concept should not usually cause many problems and can be shelved for later.

There are plenty of reasons that people might want to shelf an assignment for later (and not even that much later, mind you) if given the freedom to do so. In fact, there are plenty of people who choose to do this despite ominous warnings that “you will not learn the material and are going to fail this class if you take one day off” or the more silly “but your grade will drop.” I do not think the institution is in a place to judge these reasons, and the best reason to take a break for the day is because you need it, not because the day is 6 or 7 when taken mod 7. (There is nothing special about certain periods of 24 hours in spans of 168 hours. There is very much important about the period of time when you decide “I absolutely cannot do this anymore and am burnt out, please let me stop.”)

Of course you should learn useful stuff eventually. But a sufficiently self-motivated person knows for themselves when eventually is, and people who aren’t self-motivated really don’t care what the establishment thinks “eventually” is. (This is not to say good students want to care when school thinks “eventually” is – it’s just easier to force them to.)

Silver Lining

Fortunately some teachers do not care for this approach of “you must do exactly as I say exactly when I say.” Society (i.e. everyone except for the people involved) seems to think that there is something wrong with teachers giving students a little bit of leeway when they occasionally turn things in late with little to no explanation. But I think it is a sign of respect when teachers believe their students when they need a break. There are plenty of reasons why people can get out of “the zone” – the best thing to do is to help them get back in the zone, rather than pass judgement on them for getting out of the zone in the first place.

After all, I’d be very upset if every time I called a plumber over, I got blamed for the toilet not working.

Diminishing Returns

I immediately become apprehensive when people state they are going to do every problem in X book/Y chapter or the past Z years of the AMC 10/12. There’s a lot of reasons why this is bad – in the former case,  this is very restrictive and prevents you from moving on when you need to, and in the latter case, this seems fairly boring/stressful and is a surefire recipe for apathy. This is not to say a plan or outline for how to train is bad. Knowing what you’re going to do, even if it’s just a rough idea in your head, is better than not knowing. (The latter is still okay.) If nothing else, you’ll know how to spend your time training and won’t have to waste any of it thinking “what should I do?” while being unproductive. I recommend against having detailed plans for training because you don’t really know what you should be doing until you start getting good at it – at which point, you have more than enough of an idea of how to do it and probably should be doing something else anyways.

With this being said, two pitfalls that students commonly fall into while making/following plans of study are:

  • Being a completionist.

  • Strict adherence to “the plan.”

Let me share an anecdote. When I was in 6th grade I was just starting math competitions, was barely aware what MATHCOUNTS was, thought the AMC 8 was “the competition” of the year, and had no clue what the AIME was despite nearly qualifying for it until February or March. My preparation mostly consisted of doing random Alcumus problems with my friend during English/History class, and we occasionally did past problems from the AMCs and AIMEs and marveled at the accomplishment of making AIME (“there’s no way we’re getting into AIME”). My lack of knowledge about math competitions and my having no idea of what training was supposed to look like (“spam past tests” is probably not the right answer) meant it made some degree of sense for me to do Alcumus back then. As I learned more about the math competition scene and got my values and philosophy aligned, I gradually transitioned to using better resources.

It would make no sense and be terribly inefficient for me to do Alcumus now on the scale I did it before. I wasn’t naive enough back then to think I’d have an idea of what I’d be doing in three years, and I am not arrogant enough to think that I’ll know what I’ll be doing in three years from now. If I had to guess three years ago, my answer would’ve been embarrassingly wrong. I don’t doubt the same will happen if I guess now what I’ll be doing in three years. So the worst thing you can do is let past you dictate future you’s actions.

And it’s not just (younger) math competition kids who do this! I frequently see “regular math kids” do this when studying for a test (“I need to do all of the review problems in Chapter X!”) and it usually doesn’t turn out well. What they fail to consider is that, to put it bluntly, the review problems suck and the book they are reading from is boring and poorly written, and they need to get better problems from better books. The main driver of your studying should be you: the standard is “How much does this help?” In competition math books are “naturally selected out.” While not everything good gets the exposure it does, nearly everything that gets the exposure it does is good. This is because you cannot distort reality in competition math or in learning – you either write something that helps kids learn math or you get ignored. In school, this is not the case; standards are arbitrarily selected, and the people who pick your study material couldn’t be more detached from your situation. But it is very easy to pretend to do work when you know how to do real work; this is why I am very happy to see “normal kids” learn and be interested in math.

MAST

Hello world! I’ve been developing a math training program for AIME qualifiers who want to improve.

From the Introduction section of the website.

“The Mathematical Advancement by Self Training program (MAST) is targeted towards AIME qualifiers who want to do better on the AIME.”

I think the site will probably explain it better than this blog post could, and I still have work to do for MAST, so this post will be short. The link to the website is here.

Proof of PIE (Princple of Inclusion-Exclusion)

Introduction

Most other people’s proofs are awful to look at/so long/so boring. Mine is an easily digestible paragraph

The main difference is that others prove the statement as a whole holds inductively (boo!) while I prove that each element is counted exactly once inductively (yay!)

PIE Statement

As a refresher, PIE states that

Given sets A_1,A_2,\dots,A_n,
$|A_1\cup A_2\cup \dots \cup A_n|=\sum\limits_{i=1}^{n} (-1)^{i}\sum\limits_{\text{sym}}|A_1\cap A_2\cap \dots \cap A_i|.$

PIE Proof

Here goes.

We claim that each element is counted once.

Say that some element X is in k sets. Without loss of generality, these sets are A_1,A_2,\dots,A_k.

We proceed by induction. This is obvious for k=1.

If this is true for k, we prove this is true for k+1. For every set of sets not containing A_{k+1} with size i, there is a set of sets containing A_{k+1} with size i+1. In PIE, the sum of how many times these sets are counted is 0. There is also one additional set of sets \{A_{k+1}\}, so X is counted exactly once.

 

The Woes of Multiple Choice

Multiple choice in its current state is pretty bad. Ambiguity between answer choices can screw up a student. The reward for “process of elimination” (removing terrible answers) is usually based on luck. And if someone doesn’t know the answer and guesses… once again, luck. The way test-makers seem to want to counteract this is by 1) making every answer except the correct one completely terrible and 2) make a ton of terrible test questions (Law of Large Numbers, I suppose).

While multiple-choice is already a pretty bad format in of itself (something like the grading procedure the AIME uses is pretty easy to implement and better for large-scale automated grading), it doesn’t seem teachers will be moving away from it. So can we make multiple choice better?

Split the Point

I propose the following grading system for multiple choice be used instead:

  1. Allow students to pick as many choices as they want.
  2. If they get the answer right, they get 1/(amount of choices they picked) points.

In practice this will eliminate in-test stress. If you’ve got the answers down to 2 choices, rather than having to guess which one is right, you can choose both. It gets annoying when you have to think about one stupid question for the rest of the test because “What if I got it wrong?” and proceed to alternate between two choices.

This also rewards “partial knowledge” (knowing some but not all of the content). If you can figure out an answer or two are ridiculous, you should be rewarded for that in a consistent manner. This does exactly that.

I’ve seen teachers put “co-dependent questions” (they basically amount to being the same question) to avoid this issue and let students guess two options.

An example:

  1. What is a potato? (A) A vegetable (B) A fruit
  2. Name a vegetable. (A) Potato (B) Banana

You can assume a potato is a vegetable and a fruit and get half the points (guaranteed!) by answering 1. (B) and 2. (A). But this is annoying and cumbersome. Nobody wants to take a test like this. Why do people write tests like this?

Splitting the point would make it so you can guess (A) and (B) for the following question and not pray to the RNG gods:

What is a potato?
(A) A vegetable
(B) A fruit
(C) A Nintendo console
(D) Clothing

You can reward your students for not being an absolute moron consistently if you adopt this system.

(By the way, a potato is a vegetable.)

True/False

How would this work with true/false sections? The answer is simple: stop doing true/false. (Really.)

Fill in the Blank/Matching

Fine as is.

exploring euclidean geometry 2: electric boogaloo

First, as a teaser, I’ll publish a couple of things on here.

Preface

It is very easy to teach geometry wrong. There are plenty of high school geometry
books written for profit that do exactly this. Rarely are underlying connections pointed
out, or are thought-provoking problems required. More often than not, those who do
not enjoy math associate it with brutal calculations with ugly numbers. Rounding to the
nearest hundredth or to the nearest inch, and messing up the calculations has been a
constant source of frustration for those in typical math classes.

But the art of math is filled with beauty. In reality, most ratios are simple. A surprising
fact is that in a triangle, HG = 2GO. Most configurations will have beautiful numbers
involved, and each problem’s numbers, if any, have been chosen with care. Rarely will a
problem require unreasonable computation, because understanding the concept will
be enough to solve it.

While these theorems may seem memorization based at first, a deeper understanding
of each theorem and enough practice with them on significant, thought-provoking
problems will be more than enough to have them ingrained in your head. Putting
formulas on flashcards, writing down a formula hundreds of times, and so on are not
efficient methods because the best way to learn to solve problems is to do them!
However, if a problem does not provoke thought, it will not stick around in your brain.
Only when our brains are sufficiently stimulated will they be able to remember what
stimulated them in the first place.

Geometry can be taught wrong in so many ways, but there are so many ways to teach it
right. Only through a variety of perspectives will you understand a concept deeper.
This is one of the many resources out there, and other good resources should be used
as well.

With that said, let us begin Exploring Euclidean Geometry!

Table of Contents

A. The Fundamentals
1. The Axioms of Geometry
2. Definitions and Properties
3. Logic in Mathematics

B. Playing with Circles
1. Circles and Angles
2. Circles and Lines
3. Radical Axes

C. Triangles
1. Area of a Triangle
2. Concurrency and Collinearity
3. Lengths in a Triangle
4. Circles and Triangles

D. Trigonometry
1. Sine, Cosine, and Tangent
2. Reciprocals and Inverses
3. Trigonometric Identities
4. Graphing Trigonometric Functions

E. Analytic Geometry
1. Cartesian Coordinates
2. Conic Sections
3. The Complex Plane
4. Vectors and Matrices
5. Mass Points
6. Barycentric Coordinates
7. Trilinear Coordinates
8. Miscellaneous Algebraic Problems

F. Transformations
1. Generic Transformations
2. Homothety
3. Similitude
4. Inversion
5. Isogonal and Isotomic Conjugates
6. Projective Geometry

G. The Third Dimension
1. Volume and Surface Area
2. Lengths and Cross Sections

H. Extra
1. Pythagorean Theorem
2. Constructions
3. Directed Angles
4. Geometric Inequalities
5. Additional Problems

Progress Report?

What’s in green has already been written, and what’s in red has yet to be written. Fortunately, I only have 2 more beasts to tackle: Projective Geometry and Geometric Inequalities. Pencils/harmonic bundles should not be very hard to cover. Geometric inequalities stem from geometric equalities + inequalities, so it shouldn’t be too hard. Additional problems are for Alg/Combo/NT that I like.

So when will this be done? The short answer is I don’t know. I’m going to be busy for the fall of Freshman year ( marching band 😦 ) and it takes motivation to write the book instead of doing other things. However, I’m optimistic it could be finished in 2 months – but I don’t think I want to rush it or full effort for 2 months. The current pagecount is 306, and I’m expecting to hit at least 375. 400 might be realistic.

What? Day 3 (Beijing)

Note: Day 3, Beijing implies the third day I’ve been in Beijing.

Another note: A regular and hopefully non-boring blog post is coming up. As a note to self, things on the back-burner are (in order of priority) the following:

  1. Hard Work – How people’s idea of “You need to work hard to get really good at something” is completely wrong, and you’ll only ever get good if you don’t consider your investments “hard work.” Also talks about motivations to get top-level at certain activities and how “nice benefits from doing something” has been frequently misconstrued as “reasons to do something” (mathematics and sports are the most common victims of this).
  2. An Attempt at a Public School – Given enough money, time, and influence, it would be very easy for me to construct a good private school (or a series of outside classes or something of the sort). Most of the time when we complain about public schools all we get is an echo chamber, and I think we’ve spent so much time complaining we haven’t really been bothering to find a solution.
  3. Exploring Euclidean Geometry – A progress report and why I am so behind schedule. Estimated release times are completely inaccurate and I really hope I can get it released before I finish my freshman year of high school.

Also, remind me of this in 5 years: Mosquitoborne viruses such as malaria might be able to be solved by getting the mosquitoes to die whenever they bite a human. Whenever I take the time to learn Chem/Bio better, I might want to look into stuff that won’t affect humans but will debilitate/kill mosquitoes. If that isn’t possible, we could engineer a virus that doesn’t do anything against humans but debilitates mosquitoes (sort of the reverse of malaria).

Short explanation: Mosquitoes only get malaria from biting infected humans. This way the amount of mosquitoes with malaria can be limited and we only have to deal with the existing ones. This is probably very far off, but if in 5 years I see this and realize it is actually feasible when I have more knowledge then this is a lead worth pursuing.

Problems

Since I did 2017 AIME I #6-10 for homework a few hours before writing this post, solution outlines, thought process, and comments on these are fresh from the mind.

(Problem 6)

A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the difference between the largest and smallest possible values of $x$.

Solution:

Let y = 2x. Then we have arcs of lengths y, y, 360-2y. Let’s complementary count. The probability of no intersection is the probability both point are on the same arc, namely (\frac{y}{360})^2+(\frac{y}{360})^2+(\frac{360-2y}{360})^2. If you solve this quadratic, you see that the difference between the two values of y is 24. But we want double that so the answer is 48.

(Problem 7)
For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let T(a,b)=\binom{6}{a}\binom{6}{b}\binom{6}{a+b}. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find the remainder when $S$ is divided by $1000$.

Solution: Let c=6-(a+b). Then T(a,b)=\binom{6}{a}\binom{6}{b}\binom{6}{c}. Combinatorially, this is basically the same as saying “take 18 objects and choose a any amount of them from each group so you choose 6 in total,” which is the same as saying “18 choose 6.” Evaluating it gives a remainder of 564.

(Problem 8)

Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\angle POQ$ and $\angle POR$ are $a$ and $b$ respectively, and $\angle OQP$ and $\angle ORP$ are both right angles. The probability that $QR \leq 100$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution: These are arcs, arcs, arcs of a circle with diameter OP. The measures of the arcs 2a and 2b are from (0,150). If QR is less than 100, then |2a-2b|\leq 60 as the radius is $latex $100.$ This implies |a-b|\leq 30, and finding the probability that this happens is very standard. Using a graph, you get \frac{16}{25}, so the answer is 41.

(Problem 9)

Let $a_{10} = 10$, and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least $n > 10$ such that $a_n$ is a multiple of $99$.

Solution: Essentially you can let a_n=a_{n-1}+n, meaning a_n=n+(n-1)+\dots+11+10. This evaluates to \frac{1}{2}(n+10)(n-9). Now it is not hard to see that the answer is 45.

(Problem 10)

Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

Solution: Terrible problem.

The condition just means z must be on the circumcircle of \triangle z_1z_2z_3. So if we translate this to Cartesian Coordinates, we just want to find the x value of the highest point on the circle; this is the x value of the circumcenter, which is easy to find. The answer is 56.

Pointlessness

I believe what we did this day can be described as “hilariously pointless.”

We decide to take a taxi to the National Aviation Museum. It’s closed, and five minutes later we desperately call the same taxi driver to come pick us up as literally nobody else is there. (Fortunately, they come back and deliver us home.)

We then eat food and do more boring stuff. When we decide to go home, we walk to the further metro station. Basically, if the closer station is Station A and the further one is Station B, we walked to Station B and took the metro to Station A… when we could’ve just walked to Station A.

Our bus ride is even more idiotic; we were supposed to take Line 430 (which we end up taking), but we sit for 7 or 8 stations before realizing we are going the wrong way, which ends up taking us about 30 minutes further away from home. Oops. I would like to say we get home safely that day despite mishaps, but that wouldn’t be true; we didn’t even get home that day. When we got back, it was (technically) already the next day. Fun!