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 $w+x+y+z$.

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.

Problem 1-9: Triangle Side Expression

Problem Posts

 [Difficulty 4] [PDF]

QUESTION
(Hong Kong Team Selection Test 2009) Let $a$, $b$, $c$ be the sides of a triangle. Determine all possible values of $$\frac{a^2+b^2+c^2}{ab+bc+ac}$$

Laconic Solution Sketch
Apply Triangle Inequality, or Ravi Transformation.

Problem 1-8 Meticulous Manipulations

Problem Posts

[Difficulty 2] [PDF]

QUESTION
(2012 PMO Area Stage) If $x+y+xy=1$, where $x$, $y$ are nonzero real numbers, find the value of $$xy+\frac{1}{xy}-\frac{y}{x}-\frac{x}{y}$$

Laconic Solution Sketch
Manipulate.

Problem 1-5 Trivial Inequality

Problem Posts

Problem 1-5 (Algebra) [Difficulty 2] [PDF]

QUESTION
(Sipnayan 2011 High School Elims, Average Question 2) Suppose $x$ and $y$ are real numbers such that $$2x^2+y^2-2xy+12y+72\le0$$. What is the value of $x^2y$?

Overview/Laconic Solution Sketch
Group into squares and apply the Trivial Inequality.