# 2020 AIME II #12 video solution

I could not find a great solution to the 2020 AIME II #12, so I created a video solution myself. It is a nice geometric problem with lots of inequalities.

# MathCounts Eligibility

I receive many questions from parents about their students’ eligibility for MathCounts. MathCounts is the premier middle school math contest, the one all the AoPS kids talk about on the discussion boards. Students are allowed 3 years of eligibility to participate and once you are in high school you are ineligible. Middle school students participate no matter what math course they are studying, even if it is at an advanced level.

Depending on who you talk to, some people argue that younger students who are talented in math are allowed to participate. This may technically be true, but the reality is that MathCounts is very competitive, if not at your local Chapter, then at State or Nationals. Students spend a lot of time preparing for MathCounts. Like a lot a lot.

For some idea of how top students perform, check out a recent MathCounts National Countdown Round. Even if your student can keep up with these kids, I would not enter them in MathCounts before sixth grade. We’ve had students win at Nationals twice, but we’ve never had a middle school student three-peat there. Maybe your student can be the one. Don’t squander their personal best by participating too early.

TL;DR Don’t start your younger student in MathCounts before middle school. Use your 3 years of eligibility in 6th – 8th grades.

# LaTex for AoPS in 3 Lessons

My students who take AoPS online classes want to learn LaTex, the mathematical typesetting code. Equations are easy to read, your solutions look pretty, and for young students it can be something of an alpha move. I get them started with three basic lessons and from there they are pretty much using it independently.

Lesson 1: How to delimit LaTex.

You need to tell your environment where they should use Latex and where they shouldn’t. LaTex typically ignores spaces so if you don’t delimit it properly, you may end up with $something like this.$ At AoPS, you enclose your math with a single dollar sign at the start and end. So I ask my students to type in

$2+1=3$ This should render as $2+1=3$ Nearly every simple math expression you put between dollar signs will look nicer. Sometimes I just let them try that for a week before Lesson 2.

Lesson 2: Curly braces.

Curly braces: { } are another delimiter for LaTex arguments. I ask my student to type $3^2=9$ and this will render as expected: $3^2=9$. Then I ask them to type $2^10=1024$ and it renders as $2^10=1024$. The exponent did not format correctly because Latex didn’t know there were $2$ digits in the exponent.

So I have my students type $2^{10}=1024$ enclosing the $2$ digit exponent in curly braces. And we see: $2^{10}=1024$ as expected.

Lesson 3: Fractions

I have my student write $\frac{1}{2}$ and it renders a pretty fraction like this: $\frac{1}{2}$. This is their first example of a defined Latex command, and we use the backslash to alert LaTex to not write “frac”.

These three lessons illustrate the logic of LaTex and are usually enough to launch my students as independent LaTex users. For more Latex code, I direct them to this AoPS site or I advise them to google something like “Latex summation sign.” AoPS also offers a test site where students can practice their code.

# Math Kangaroo vs. MOEMS

Parents often ask me about math contests for elementary students. The two heavyweights are Math Kangaroo and MOEMS:

The popularity for MK is a head-scratcher for me. I do think it’s a great exam and students can learn a lot from studying past contests. The questions are a bit more “puzzley” in nature than what I find on the AMC.

Working against MK is that it’s an annual exam; once and you’re done, then you go home. It lasts over an hour, which is a long time for most first graders. You can purchase their old exams, but only individually, the solutions cost extra, and the online store is awkward to navigate.

MOEMS stands for Math Olympiad for Elementary and Middle Schools. I have my students take the exam, turn in their papers, and then immediately following we go through each question one at a time. I write the problem on the board, and ask for volunteers to explain how their solved the problem. While they describe their solution I scribe for them while also helping them with their newly learned mathematical vocabulary. For example a student may say, “I connect those 2 points on the left.” I may gently say, “Oh, you mean you draw a segment connecting vertices A and B?”

I continue by asking if there are any alternative solutions. Students learn from each other, which makes the problems seem within their grasp. Over the course of a season, the students build friendships that lead naturally to higher level math contests like MathCounts and ARML.

Old exams with solutions are published in books that can be easily found on the MOEMS website or Amazon. You can participate officially by paying a registration fee, and at the end of the season they provide you with trophies, pins, and certificates. Or you can use the old exams unofficially in small groups or individually, and on your own schedule and rules.

The word “Olympiad” in the name conjures the International Mathematical Olympiad for high school students, but there is no relation between the two math contests. In fact, it’s a friendly collegial math activity that’s perfect for our younger math students.

My own affection for MOEMS stems from my personal experience with my homeschooled daughters. Over several seasons they befriended other mathy kids while I befriended their parents, making it easy to recruit team members for MathCounts and other activities. MK is homeschool friendly, and I would put MOEMS as homeschool agnostic, not exactly welcoming, but easy to work with.

Section 8.2 of AoPS Introduction to Algebra introduces (i) the idea that a linear equation looks like $Ax+By=C$  and (ii) the slope of the line.  With some students I get in a pinch because sometimes they are already somewhat familiar with graphing so in an effort to be efficient we rush into the definition of slope and then subsequently backtrack because of confusion.

So this week I was prepared for my student.

We work the problem where we first calculate the slope of a line (Problem 8.7), using $2$ pairs of points to get the idea that the slope is the same everywhere on the line, no matter which $2$ points you choose.  But I was more careful.  Before getting into $\frac{y_1-y_2}{x_1-x_2}$  I give him a heads up.  “Now I know this formula is going to look like it’s coming from outer space, but just bear with me for a second while we do these calculations.”  Then we calculate this unnamed ratio for 2 pairs of points and we find they have the same result, $\frac{1}{2}$.  Then I ask him:

“What do you think?  Do you think this formula will be $\frac{1}{2}$ no matter which $2$ points we choose?”

Thoughtful silence.  We had carefully traced the path of  Hopsalot the rabbit, so I reminded that between every consecutive point, we go right $2$ units and up $1$ unit.  Eyes light up.  He gets it, that the ratio is always going to be the same.

Even better, we didn’t actually finish the problem, or name that this ratio is called slope.  I like leaving him with this epiphany for a while so he can internalize it a bit.  Then we can review this once more before proceeding.

Since we use the slope to generate the point-slope form of the line, it’s imperative that students understand that the slope is the same everywhere on the line, no matter which 2 points we choose.  We use this property to generate the equation because it’s one of the few things that consistently describes every point on the line.    I’m going to give him this heads up and reiterate as much as possible that the slope is an inherent property of a line, no matter which 2 points you choose.

# Mistakes I Do and Don’t Correct

Math Mistakes I Will Not Tolerate:  Stream of Consciousness
Some students use the equals sign to mean “…and then…”  As in, I have $12$  apples and then someone takes away $3$ and then I’m left with $9$ and then someone gives me $4$ more apples and then I have $13$ would look like this:

Now we have an illogical statement where on the left we have $12-3$ and on the right we see that it is equal to $13$.  I think this is part of the reason so many students find math frustrating: they see illogical statements like this one and think math is a subject where rules can be made and broken at whim.  I am quick to correct errors like this.

Math Mistakes I Allow:  Inefficient Exploration
My students who are new to algebra will not always choose the most efficient solution.  For example, when solving this equation for x:

an experienced student will first subtract $3$ from both sides and then divide both sides by $5$.  Some of my less experienced students have more creative ways of solving the equation.  They may choose to first divide by sides by $5$, yielding

Or they will begin by subtracting $5x$ from both sides or any number of unusual attempts to isolate $x$.  My mantra is “As long as it’s legal, we can try it.”  The most fundamental thing a student needs to learn about algebra is the idea that when you do the same thing to both sides of an equation, you still have equality of both sides.  This is my main focus, that if they do something to the left hand side of the equation, they must do the same thing to the right side.  If the cost is a little extra time spent in going down some inefficient rabbit holes, so be it.

The other reason I tolerate Inefficient Exploration is that it won’t always be obvious how to solve a problem, until you try to solve it.  Many students when they encounter a problem they don’t know how to solve will freeze.  They may have some ideas, but since they don’t know if it will work, they don’t take the risk of trying.  I want my students to be like the one who decides to divide by $5$ first, just to see if it will work.  It might, it might not, but we’ll never know unless we try.

# AoPS Options: 4 Ways to Study!

Many parents contact me because they are confused by the options for studying AoPS curriculum. And it is confusing! Here is a summary of options with my list of advantages and disadvantages:

(1)  AoPS textbook only.

Advantages: go at your own pace, more practice problems, if parent teaches it is least expensive option

Disadvantages: no group/community learning, may need to hire a tutor (becoming most expensive option)

(2)  AoPS online class.

Advantages: student is on a set schedule, i.e., you know exactly when you will be finished with a course; student joins the AoPS online community of math loving students, opportunity to learn and practice Latex skills, writing problems evaluated by human teacher.

Disadvantages: students often do not take deadlines seriously and do not complete homework problems by deadline (homework extensions only compound this problem), little flexibility for family schedules, some students do not like text-based class, fewer homework problems.

(3) AoPS self-paced class.  (Prealgebra only)

Advantages:  very similar to online class without online classroom, but with great flexibility in timing

Disadvantages:  students historically tend to not complete the class, access limited to 9 months per half course.

Advantages: In person classes with online video during pandemic

I support students who are using all these options with an online whiteboard and videoconferencing.
For example with the textbook only option (1), I work through the problems with the students and then assign the Exercises, Review Problems and Challenge Problems as homework.
I also support students who are taking online classes (options 2,3,4) with light homework help.  Sometimes I remind them of key points from lecture, a short prompt here and there to get them started.  I also model solid mathematical documentation on the whiteboard to help them avoid errors and improve their writing problem responses.  I also get them started with Latex (more exciting than it sounds!).

# My 5 Types of Students

In the spirit of market segmentation, I’ve identified the 5 types of families who hire me:

1. Math-Maturity-Mismatch Students  These are the younger students who have high math ability, but they have the maturity of a typical kid, and they have trouble following through on their homework independently.  Students meet with me once or twice a week for “homework time” saving parents from nagging their kids so they can have a better relationship, and the students have more fun.
2. Socratic-Scaffolding Students  These students blank when they see a problem, and all they lack is the tiniest hint to get them started.  “Hm, today’s topic was triangle inequality.  Can you tell me the rule for triangle inequality?”  And suddenly, just like that, they are off to the races.
3. “I Must Qualify for Nationals” Students  These are the students who are keen to level up in MathCounts, qualify for AIME, or just generally improve their math contest scores with an eye on college admissions.
4. “You Look Like Me” Students  Yes, female students like that I’m a girl, and I really get them.  There’s something to be said for single-gender math education, especially in the critical middle school years.
5. Last Comic Standing Students  These students like me for my nerdy sense of humor.  And most give it as well as they get it.  If I use my standard “Problems involving FRUSTRums are very FRUSTRating” joke, they respond with their favorite one-liner:  “y=mx+b.”

# What Everyone Should Know About Math Contests

Math contests encourage a growth mindset.

Students who are accustomed to earning 95% or 100% on their worksheets and test papers understandably view that as a badge of honor, but the desire to maintain high grades can lead to wanting to “look smart” rather than “be smart.” This “tyranny of 100%” can encourage students to avoid challenge for fear of a lower grade. Because math contests are designed to identify only one winning student, even accomplished math students may earn far less than 90%. Regular participation in math contests that don’t impact GPA and where lower grades are common enables students to relax and learn for the sake of learning and not in order to “look smart.”

With academic support, math contests can replace a math curriculum.

I’ve had students who refuse to follow a traditional math curriculum, but were happy to learn their algebra, geometry, and discrete math in 3 years of preparing for MathCounts. With a strong coach who can explain and derive the concepts behind each problem, students can learn with greater retention. They don’t learn the Pythagorean Theorem once and then forget it. They are forced to deploy it in novel situations week after week at each MathCounts practice.

Math contests don’t always reward speed or math tricks.

Most modern math contests do not ask students to solve the identical multiplication problems in 10 minutes. Instead they reward thoughtful persistence and grit toward arriving at a solution. Each problem is different from the one previous and the one following. A geometry problem using similar triangles will be followed by a probability problem involving dice will be followed by a problem solved by modular arithmetic.

Math contests encourage academic bravery and risk-taking.

Often it isn’t obvious whether a particular strategy will result in a solved problem. It’s like navigating your way to the drugstore without a map by peering down a street and considering whether it might be there. A particular street may or may not have the drugstore, but you won’t know unless you take a risk and travel down the street. Likewise, in math, the most interesting problems don’t have a map showing you the way and often the student needs to take a risk and try something to know if it works.

Make math contests work for you

Many parents and students believe math contests aren’t for them because of the above reasons, but don’t let these misconceptions steer you away from this useful tool.

I’m your expert in all things related to Art of Problem Solving math curriculum and math contests generally.   I guide students who are taking the online AoPS courses and studying from the textbooks independently.  I help students prepare for AMCs and I coach a homeschooled MathCounts team every year.

You can also call me a Richard Rusczyk fan girl.  One of my favorite talks is one he delivered at Math Prize for Girls in 2009, when he showed this slide:

“I certainly wish your website and materials existed when I was in high school. I went through junior high and high school without ever missing a question on a math test, and then took [Math] 103 and 104 at Princeton, which was one of the most unpleasant and bewildering experiences of my life and poisoned me on math for years.”
–Princeton University alum
He continues:

“I want you to think for a minute what this student’s middle school and high school teachers thought when he went off to Princeton.  They thought, “We succeeded.  He went off to Princeton; we’re awesome.”  They never saw this.  I’m sure he didn’t go back to his middle school teachers and say, “Yeah what’s up?!?  You didn’t prepare me for this.”

“So they didn’t get this feedback, and this happens a lot.  Kids go through school, some very good schools, they get perfect scores on everything, and then they come to a place like MIT, a place like Princeton, they walk into that first year math class, and they see something they’ve never seen before: problems they don’t know how to solve.  And they completely freak out.  And that’s a bad time to have these first experiences.  Having to overcome initial failure.”

Don’t let this happen to your student.  Front load their math education by challenging them early in their academic career.  I’m talking elementary and middle school.