Problem Post 2-6: How Many Shapes, Part 3: Rectangles in Not-Rectangles

		Problem Posts
		Part 3 of 4 of How Many Shapes?

[Level 2-3]

While it would be far more fun for me if I jumped from topic to topic, I believe I owe it to you, dear readers, to flog this horse until it's brain dead. Completeness is a virtue.

So last time we finished off the case where we count rectangles in rectangles, even when some 'matchsticks' or grid wires are missing. Now onwards to rectangles in non-rectangles.

CAN WE DO BETTER?

• What if the original grid were a rectangle, instead of a square? Or an irregular figure?
• What if we were counting rectangles instead of squares?
• What if some of the grid 'wires' are missing?
• What if we're working on a triangular grid, counting triangles? Does our logic still apply?

2 Rectangles in Non-Rectangles

This is ad hoc land. The base technique would again be to count one-by-one, but as we will see there are tricks for certain special irregular figures.

Problem Post 2-5: How many Shapes, Part 2: Rectangles in Rectangles

		Problem Posts
		Part 2 of 4 of How Many Shapes?

[Level 2-3]

So in my previous Miscellaneous post, we solved a fairly simple Internet puzzle, counting the number of squares in a square grid. But now, we ask ourselves if we can do more:

CAN WE DO BETTER?

• What if the original grid were a rectangle, instead of a square? Or an irregular figure?
• What if we were counting rectangles instead of squares?
• What if some of the grid 'wires' are missing?
• What if we're working on a triangular grid, counting triangles? Does our logic still apply?

Problem Post 2-4: On the Escalation of P*cking Problems

Problem Posts

Image Credit: "Cardboard Box City" by James Nash. Licensed under CC BY-NC-SA 2.0.

No, I don't think they'll make you solve this problem in fifteen seconds mentally.

[Level 1]

\( \def \cm{\rm{\,cm}} \def \mtr{\rm{\,m}} \)

You may wonder why I used the P-word for something as "simple" as packing boxes. I mean, the question is straightforward... right?

THE PROBLEM
You have a big rectangular pallet upon which you must place identical rectangular boxes. You can't stack, collapse, or overlap boxes. The sides of the boxes have to parallel to one of the sides of the pallet. What's the maximum number of boxes you can place?

It's a familiar but unremarkable question, especially for many mathletes who've had to do it under time pressure. (Fifteen seconds mental? Easy!) And the real-life applications are obvious - putting things into shipping containers, for example.

Miscellaneous 2-1: How many Shapes, Part 1: How many Squares?

Part 1 of 4 of How Many Shapes?

[Level 0]

I can't really classify this a Problem Post, because its roots lie more in Internet culture than in math contests.

Remember this nasty little thing making the rounds on social media?

Once you read this post, 100% will put this puzzle behind us.

We could just manually count squares, but that's boring, and we want to find a method to do this for huge squares.

PROBLEM 1
Devise a method to rapidly solve problems of this type, no matter the size of the square grid.

Problem Post 2-3: EROs Should be Taught in High School

Problem Posts

[Level 1-2]

Disclaimer: I'm using this problem to prove a point, that is, that it's feasible, and easy to teach elementary row operations, (in particular in conjunction with Gauss-Jordan Elimination) at the high school level (at the very least for contest-involved students). While central to linear algebra, Gaussian elimination can be appreciated at a fundamental capacity. Essentially it's just a way of manipulating numbers, no different from right-to-left addition or long division.

The main point, I believe, in introducing students to GJ, is to demonstrate that any system of $n$ linear equations and $n$ unknowns can be solved (i.e. all solutions, one or infinite, found) quite uncreatively.

If you want to know how to solve the problem, the solution is at the bottom of the article.

QUESTION
(16th PMO Orals) Suppose that $w+4x+9y+16z=6$, $4w+9x+16y+25z=7$, $9w+16x+25y+36z=12$. Find $w+x+y+z$.

Side Story 2-1: Do You Construct, or Do You Freehand?

Newton by William Blake. Public domain.

By using only compass and straightedge, you become part of an ancient tradition.

Quick Question:
Do you prefer using a compass and a straightedge to draw geometric diagrams, or would you rather freehand?

Most of the people I know freehand. I would posit that the simple reason is that geometric construction isn't a topic touched upon at great length in most of our classrooms, and dabbling with the compass is mostly done in drafting class (that is, if you had one; I didn't.). When it comes, however, to people who deal with more elaborate [Difficulty 4+ in this blog] geometry problems on a regular basis, the question does become a tad more relevant. The figures get quite complicated, and for some, the best way to make any sense out of anything is to draw them as precisely as possible. Hence constructing.

Video: Sum of Consecutive Squares

[Level 0]
Some things really are easier shown than said. Here's an uber-concise graphical 'proof' of the famous identity
\[
\sum_{i=1}^{n}i^{2}=\dfrac{n\left(n+1\right)\left(2n+1\right)}{6}
\]
(For those unfamiliar with \(\Sigma\) notation, here's a quick explanation. Trust me, it's not as scary as it looks!)

The High School Gap

See you on the other side.

DISCLAIMER

While I’ve observed this phenomenon independently early on, I cannot take credit for being the first to point it out. I first heard the version I will be elaborating on in Mar del Plata, July 2012. The diagrams, further explanation, and tips below, however, are my original work.

I am not (yet) an expert in mathematics education, and the explanations and diagrams below come from my personal observations and not from any formal study. If you want to subject this phenomenon to research, however, do consider e-mailing me. I’d be happy to help!

Problem Post 2-2: A Sampler of `Olympiad' Geometry Concepts

Problem Posts

[Level 4]
Henry

This post is meant to be a `sampler' of sorts, to show the most common tag words one will see in olympiad geometry problems.

I've been fortunate enough to find two remarkable problems, and solve them in ways that form a whirlwind tour of the subject. The first problem demonstrates side chasing, isosceles triangles, some cyclic quadrilaterals and spiral similarity. The second problem demonstrates power theorems, cyclic quadrilaterals, collinearity, and triangle geometry.

Problem 1

This one comes from Andreescu and Gelca's Mathematical Olympiad Challenges.

QUESTION
(Andreescu, Gelca) Let $B$ and $C$ be the endpoints and $A$ the midpoint of a semicircle. Let $M$ be a point on the line segment $AC$, and $P$, $Q$ the feet of the perpendiculars from $A$ and $C$ to the line $BM$, respectively. Prove that $BP=PQ+QC$.

Mathletes’ Greatest Secrets Finally Revealed Episode 6: Mathematics Olympiad Summer Camp

Mathletes’ Greatest Secrets Finally Revealed

Episode 6: Mathematics Olympiad Summer Camp
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Henry

Screenshot from imo-official.com

If you make it all the way, your name ends up on this site. For better or for worse. Forever.

Each year the Mathematics Olympiad Summer Camp selects and trains the Philippine Team for the International Mathematical Olympiad.

If you’re a high school senior, this will probably the last training-contest circuit you will be doing before entering university. Everybody else in your batch will have hung the gloves before you do. Know that whatever happens, you’ve been given the rare honor of staying in the ‘Math Games’ until the very end.

Mathletes’ Greatest Secrets Finally Revealed Episode 5: Philippine Mathematical Olympiad

Mathletes’ Greatest Secrets Finally Revealed

Episode 5: Philippine Mathematical Olympiad
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Henry

Image Credit: "weird pi sculpture at the lavender farm" by Brian Ellin. Licensed under CC BY-NC-SA 2.0.

The letter pi is a apt motif that recurs in each yearly iteration of the PMO logo.

The Philippine Mathematical Olympiad is the premier mathematics contest for high school students in the country. Among the contests and programs we’ve discussed, only this one bears the legitimacy of being the country’s national Olympiad.

Mathletes’ Greatest Secrets Finally Revealed Episode 4: Mathematics Trainers' Guild

Mathletes’ Greatest Secrets Finally Revealed

Episode 4: Mathematics Trainers' Guild

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Henry

(Image credit: publicdomainpictures.net)

YMIITP often brings participants to interesting locales around the country.

The Mathematics Trainers’ Guild offers one of the largest scale training programs in the country, and sends participants to a large variety of international events. It also keeps quite a high profile – if you perform, you will likely see your name in the papers.

Problem Post 2-1: Factor Sums, and the Distributive Law

Problem Posts

[Difficulty 1; some parts Difficulty 4]

Who knew power series multiplication could help you at the grocer? (see Question 2)

Just a quick post for younger readers. Often one will find oneself using `brute force' approaches when obvious tricks and shortcuts exist. Most of the time, this is justified -- many tricks are usually too arcane to remember, or too impracticable to execute realistically. This is neither. It's fast, simple, and it could save you in a Do-Or-Die (I've used it before; our team won!)

What is a factor?

First things first, right? Not everybody defines the word “factor” the same way. I will be using the convention used in most local contests: A factor of an integer $n$ is a positive integer $a$ for which there exists an integer $b$  such that $n=ab$. So by our convention $4$ and $-4$ both have exactly three factors: 1, 2, and 4. We do not consider $-2$ as a factor, even if it divides both $4$ and $-4$. On the other hand, a factor of a number is a proper factor iff it is not the number itself. Often 1 is also not considered a proper factor. With this ambiguity, however, the term is not used too often in contests.

Side Story 2-1: Remastered - Eight Signs You're A Maturing Mathlete

Note: Part of this Side Story came out back in 2013 when this blog was still named Solvespace. Here's an updated version, with lots more content!

So you're not anymore that newbie who writes full solutions on scratch paper in case it's graded. But have you become a Matured Mathlete?

1. You retain a fetish for $\mathrm{\LaTeX}$.

Microsoft Equation has become superb and easy to use, but nothing can really beat the oomph of sexily typeset mathematics (or chemistry, or physics):

$$ i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)$$Look at my pretty equation, ye mighty, and despair!

(Sorry, Shelley.) Back in the day, to typeset in $\mathrm{\LaTeX}$ was to be the Voice of Unerring Authority. Today, well, it's still pretty (for me, at least). And it still commands a sort of respect from the reader (Me, at least. But who can't appreciate good vector renderings?) Sure, I might just be saying $\int e^x = f_u\left(n\right)$, but the typesetting nevertheless screams, "I am serious about my math". (But then I wouldn't be; I missed a differential $\mathrm{d}x$ over there.)

Mathletes’ Greatest Secrets Finally Revealed Episode 3: Metrobank-MTAP-DepEd Math Challenge

Mathletes’ Greatest Secrets Finally Revealed

Episode 3: Metrobank-MTAP-DepEd Math Challenge
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[PDF/Printable]

Henry

"Redu CS3aJPG" by Jean-Pol GRANDMONT - travail personnel. Licensed under CC BY 3.0 via Wikimedia Commons.

Tempus fugit: Time flies, and so must you; nowhere else is speed and accuracy as vital.

The Metrobank-MTAP-DepEd Math Challenge, colloquially referred to as MTAP (the Mathematics Teachers’ Association of the Philippines, which organizes the event) or MMC, is one of the most venerable and well publicized contests in the country; it has been around in one form or the other since the 70’s. It is also where many math people start out. Unlike other contests, MMC is open to students from the very first year of primary school.

Mathletes’ Greatest Secrets Finally Revealed Episode 2: School and University Hosted Contests

Mathletes’ Greatest Secrets Finally Revealed

Episode 2: School and University Hosted Contests
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[PDF/Printable] coming soon!

Henry

(Image credit: P. Cowie. Public Domain.)

Tip #1: Keep a formula notebook (see below). You'll find it useful, and so will archaeologists 3500 years later.

I will be discussing this sort of contest (e.g. Sipnayan, Lord of the Math, et al) at once, as there are so many and doing them one at a time would needlessly prolong the series and bore you, my precious readers. There are also other contests that contain math among many subjects; if you’re performing well in the math-only contests, the math there should be manageable.

Back to math-only contests. These are usually organized by a student organization within the school, with varying levels of support from the resident math department.

Again as an NCR resident, most of my examples will be NCR contests. Contests elsewhere of this nature should hopefully not be too different.

Mathletes’ Greatest Secrets Finally Revealed Episode 1: Eight Myths About Mathletes

Mathletes’ Greatest Secrets Finally Revealed

Episode 1: Myth Slaying, or Eight Myths About Mathletes

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Henry

(Image credit: Cornelis Cort. Public domain.)

Don't let new myths pop up where old ones die. Vanquish some myths today!

Before we tackle some actual contests, I’d like to dispel some common myths about competing in mathematics.

Myth #1

In a word: no. Hard work and true determination can more than take the place of talent.

Myth #2

We really can’t tell. Many questions in competitive mathematics go beyond standard exercises and speed drills. I’d say it’s safer to subscribe to a competitive math program, or get some books on the topic. At any rate, your obscenely fast arithmetic will be an advantage anywhere.

Mathletes’ Greatest Secrets Finally Revealed Episode 0: Introduction - Why Compete?

Mathletes’ Greatest Secrets Finally Revealed

Episode 0: Introduction - Why Compete? 
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Henry

"Everest North Face toward Base Camp Tibet Luca Galuzzi 2006" by Luca Galuzzi (Lucag) - Photo taken by (Luca Galuzzi) * http://www.galuzzi.it. Licensed under CC BY-SA 2.5 via Wikimedia Commons.

To echo a famous mountaineer: it's there -- will you climb?

Preliminaries

If you’re reading this, one of these things has happened:

• You want to be the very best (in math), that no one ever was!
• You’ve been admitted into your school math varsity and you want to know what you got yourself into.
• You’ve accumulated experience in a few contests, but you’re stuck in a rut and you don’t know what the next step is.