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.