# Book – Proofs in Competition Math

This is a book which is about proofs in competition math. This blog post is a promotional one for its release.

Authors: Alex Toller, Freya Edholm, Dennis Chen.

# Important Info

Preorders start on March 14th, also known as Pi Day. Yes, this was on purpose. (3/14)

Release is on April 5th. The best way to remember this is you will have to run for your life if you don’t get this book on four-five(4/5)

# Why you should pre-order (when the time comes)

First, if you want to see my writing… this book doesn’t have much of it at the time of this post. (Oops.) Unfortunately, I’ve been bogged by other obligations, so I haven’t written much. (If you think the issue of splitting the earnings and me potentially getting money for nothing is an issue, I do too. More on that later.)

But if you agree with what I say on this blog, here’s something you’ll definitely agree with when the book comes out (and even if you don’t agree with me most of the time, you will agree here) – this book is high-quality. I will have proof for that (aka samples) soon, if the other two agree to release nontrivial yet non-significant portions. I’m writing Inversion, so if you liked EEG’s Inversion, you are going to like this one even more.

Also, Alex and Freya are very reputable within the math community themselves (though  I suspect most readers of my blog already know this). (In fact, they’re much more reputable than me, but shh!)

Anyway, this is quality, this is hype, and this covers a lot of stuff.

# Actual content

Why you should pre-order: Part 2.

Here’s a rough ToC, with wording that is totally inaccurate.

Part 1: Proofs (this is logic in general, also stuff like iff. Good for beginners.)

Part 2: Algebra

A: algebra for noobs

B: Basic Inequalities + Complex Numbers + just stuff you should know for polynomials

C: Really hard stuff (see newton sums)

Part 3: Geometry

A: normal stuff

B: normal, but more advanced stuff

C: really hard normal stuff + bary/polar/cylindrical + inversion (I’ll get spiral similarity and homothety added if I can)

Part 4: Number Theory/Combinatorics

A: intro+interesting but unimportant stuff (its kind of the equivalent to spiral similarity or inversion)

B: More interesting but unimportant stuff.

C: Classic NT (Bases, mod arith)

D: Hard NT (HELP)

E: Even harder NT. (HELP II)

Part 5: Open Problems + for fun

A: Goes over open problems. Makes no progress but defines the backgrounds. (If we could make significant progress, we’d submit that as an article hm?)

B: Pythagorean’s Proofs (my favorite) and Fake-Proofs (my LEAST favorite).

# I’m not being paid for this

I forget if I’ve said this to Alex, but I don’t intend to be paid for this simply due to the fact I’ve done an embarrassingly small amount of work on the book. I will still

a) take responsibility for the final product

b) be involved in marketing

c) actually work on the book now

but the little work I did does not warrant payment. (It’s a wonder I’m on the authors list…) I intend to make up for that by working on it now.

# In Conclusion…

Buy the book when it comes out!

I’m a lazy bum too, so I deserve all the blame for flaws because I’d be able to fix them if I was paying more attention and none of the credit for success because I didn’t do anything. (Oops.)

Also, I don’t know where Homothety/Spiral Similarity will go. Preferably next to Inversion, with Homothety before Spiral (since Homothety is a special case of Spiral).

Please support this by sharing with your friends or whoever might be interested!

(If this post seems lazy, it’s because I’d rather get onto writing the book.)

Edit: I’m going to put this in every category, so people see this. I also will update my website soon.

# Quick Update on IGP 2

Hello world,

I haven’t been posting in a while.

We may have another IGP soon. However, we’re also planning to make an entrance exam for next year’s kids (more details on this when we’ve actually started planning some things), so any new problems we create will probably go on the test. However, I usually create some new problems for every IGP, so I’ll probably be using my new problems for IGP and for the Entrance Exam. This causes an issue because I can’t post the IGPs up on geometryexplorer.xyz because I don’t want the exam questions to be findable, even if they aren’t explicitly marked. (For my kids, they will be on the IGP Handouts, if I decide to do it next Tuesday.)

Because of this, the IGPs from now may not be public or only a selection of the IGPs will be public from now on. I apologize for this, but I think the Entrance Exam will be worth the wait.

(P.S. To my kids: Please start working on writing problems! Most of you are very far behind, unless you’ve been working behind my backs, which I hope is the case right now…)

# An Angle Problem in MPP

Today for class I gave a problem that appeared as #21 in a Mock AMC 10 this year.

The problem is available here.

Non-spoiler comments follow below. However, I still recommend you try the problem before reading if you’re interested.

When I assigned it, I thought that this problem was very misplaced for its difficulty; it is literally just a bunch of given angles with a nice observation and some algebra. Nevertheless, it was a good problem, which is why I gave it.

I still think it was misplaced, but now I think it was misplaced in the opposite direction.

Let’s take a look at 2017 AMC 10A #21, which is recent, placed in the same location, and a geometry problem following the same vein (looking at angles). However, this one is very straightforward. The crucial observation could be made almost instantly, and the application is straightforward. The crucial observation in the mock is much more difficult to make. Though the way the problem is written gives a hint to the crucial step, some cleverness in algebraic manipulations is required to actually find the answer.

Even though it’s a tad too hard for the test, and none of my kids managed to solve it in 30 minutes, I think they’ll enjoy the problem and its solution, which in my eyes was pretty clever.

(Clarification: The above post refers to the mock problem.)

# Pushups or Homework?

I assigned kids in my class homework. Most kids are usually good about homework, but some kids don’t care and never do it at all.

Recently, I decided to make kids who don’t do homework do pushups instead. We have 8 kids in our class, and 5 problems for homework. Let’s take a look at my homework policy:

“If you do your homework, you get 5 pushups for every problem you get wrong.

If you do not do your homework, you get 2 pushups for every problem other kids get right, and 5 pushups for each extra credit problem the class does (if two people from the class do #34 independently or with each other, it counts as 1 problem regardless…)”

Also, I am considering making the minimum # of pushups for people who don’t do homework be what you would’ve got if you submitted them all wrong.

Let’s see who doesn’t do homework now.

# IGP

Now that I’ve actually held an IGP class, I will offer a couple of comments on IGP.

1. If you’re trying to surprise your students, make their handouts have the same amount of questions. (Have them work alone.) Kids tend to talk a lot, even if you tell them not to. Someone will say “There’s only 5 problems!” and someone else will say “Wait, isn’t there 6?” I tried to lie through this, but it probably wasn’t very convincing…
2. I’m not a student, so I don’t get the sense of wonder that the students might get from seeing who has the same problems, the whole handout of problems, and can I please just have my problems explained teacher? This will probably apply for someone else too. Nevertheless it was really fun watching them discuss.
3. I think I made the handouts too hard. Students majorly sucked (compared to how I expected them to do) on the handouts even though I gave formulas. I didn’t expect them to telescope so well though, which is a plus for me…

I feel it is important to reiterate that IGP is not a teaching method, it is a review method!

Handouts for IGP (including individual ones) can be found on geometryexplorer.xyz/igp.

# GG Garage Groupies!

MATHCOUNTS Club tryouts begin on 9/14. We went over 2017 School Sprint Round for MATHCOUNTS (you can probably pirate this somewhere), then we began Intro to Trig. For anyone looking to catch up on what we taught today, see here for basic trig, and see here for the entire trig unit. We’re going to be doing Complex Numbers, so see here for complex numbers, which is very related to trig.

# Triangle Centers

We introduced the “four common lines” (altitudes, medians, angle bisectors, perpendicular bisectors) and the “four centers.”

For any of my students who forgot,

Angle bisectors -> Incenter (I)
Medians -> Centroid (G)
Altitudes -> Orthocenter (H)
Perpendicular Bisectors -> Circumcenter (O)

Proof main ideas:

Incenter – Let two of them intersect at I, drop perpendiculars, congruence chase.
Medians – Trivial by Ceva’s (the proof isn’t that informative)
Perpendicular Bisectors -> Let two of them intersect at O, use the fact that AO=BO and AO=CO to get AO=BO=CO. (This is assuming the B/C perp bisectors are the ones that we let intersect at O; it doesn’t really matter.)
Altitudes – Medial triangle + perp bisectors (let them be the perp bisectors of another triangle)

Corollary: Incenter is equidistant from sides.
Corollary: Circumcenter is equidistant from vertices.
Corollary: The 6 centroid triangles (the small triangles made by drawing the three medians) have the same area.

We also went over the proof of O/H isogonal conjugates. This was a simple right triangle angle chase.

The handout/problems are here and here, respectively.