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 $BP=PQ+QC$.

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 1-6 Romance of a Right Triangle

Problem Posts

[Difficulty 4] [PDF]

QUESTION
(Canadian Mathematical Olympiad 2013, Problem 3) Let $G$ be the centroid of a right-angled triangle $ABC$ with $\angle BCA = 90^{\circ}$. Let $P$ be the point on ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circumcircles of triangles $AQG$ and $BPG$ meet at a point on side $AB$.

Laconic Solution Sketch.
Using similar triangles, we can show that this common point is the foot of the altitude on $AB$ from $C$.