,
,
,
,
or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
,
isn't even rigorously defined in this expression. What exactly do we mean by 
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that 
,
or ![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](http://latex.artofproblemsolving.com/2/6/0/260f97a1cf1ea8722ff97a29b83bac7bfe83e577.png)
etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function 
given by the mapping 
is undefined for 
.
is undefined because dividing by ![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](http://latex.artofproblemsolving.com/9/9/3/993c78b62ec2c84b5d77daa8e9a8cd6f70fcf904.png)
So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let 
for each 
which means that via this order topology each subset has an infimum and supremum and 
is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In ![\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}](http://latex.artofproblemsolving.com/6/d/8/6d8fc1eca5c791d430a7168cd3b37a02104fa318.png)
So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](http://latex.artofproblemsolving.com/a/0/b/a0b67076efcc014c78b2023be1151c84b57c1503.png)
So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define 
as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by ![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](http://latex.artofproblemsolving.com/8/4/e/84ed3d6ab237df970333b87beaef5311caa46185.png)
which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, 
means complex infinity, since we are in the complex plane now. Here's the catch: division by ![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and 
Furthermore, we actually have some nice properties with multiplication that we didn't have before. In 
it holds that![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](http://latex.artofproblemsolving.com/8/e/0/8e035b4841d20c5ad671c46355cc667ca01e533b.png)
but 
and 
are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with ![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by 
be positive real numbers such that 
Find the minimum value of 
Prove that
and 
connect the points 
and 
These arcs lie in the same half-plane defined by line 
in such a way that arc 
lies between the arcs 
and 
Point 
lies on the segment 
Let 
,
be three rays starting at 
lying in the same half-plane, 
being between 
and 
For 
denote by 
the point of intersection of 
and 
(see the Figure below). Denote by 
the curved quadrilateral, whose sides are the segments 
and arcs 
and 
We say that this quadrilateral is 
if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals 
are circumscribed, then the curved quadrilateral 
is circumscribed, too.
Prove that
red marbles, 
yellow marbles, 
green marbles, and 
blue marbles. If Amanda takes the marbles one by one without replacing them until the 
marble. Then the probability that the 
marble is red is?
and 
that satisfy
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
lies on diagonal 
.
.
has 
,
,
.
.
indicates my favorites.
is defined over the positive integers as follows: 
,
for 
prime, and for all relatively prime positive integers 
.
is the smallest positive integer such that 
,
,
,
,
,
,
?
and 
be the greatest odd divisor of 
for even 
denote the product of every odd positive integer less than 
for positive integers 
and 
where 
is minimized, find the number of divisors of 
.
consists of a line and a point not on the line. The distance from the point to the line can be expressed as 
,
.
.
are positive real numbers such that 
,
,
,
be a convex pentagon satisfying 
,
,
.
be the intersection of lines 
and 
.
has a perimeter of 
and an area of 
,
be an acute scalene triangle with longest side 
be the circumcenter of 
are chosen on 
and 
.
,
,
,
.
such that 
is divisible by 
by 
,
such that in every row and column, the number of 
or 
is valid, while a row with squares labeled 
or 
is not valid.) Find the remainder when 
.
,
be real numbers satisfying the system of equations
What is the value of 
?
at 
and 
are 
be the set of all 
,
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For my 999th post, over the past few months, I've been slowly compiling a handout briefly covering Triangle Configurations in Olympiad Geometry, or "American" Triangle Configurations in Olympiad Geometry, meant for everyone who would like to learn about triangle geometry in an olympiad setting, and the background knowledge being EGMO.
Olympiad geometry is a subject in which there is a significant amount gained from just being familiar with the configurations. We see AoPSers on this forum talk about things like the "Ex Point", "Iran Lemma", "Humpty and Dumpty Points", "'Median-Incircle Concurrency", etc. What are these? What are the proofs of these things? What are problems that include these? How can I learn the power to insta-kill old problems by citing some "well known" lemma?
In this handout I have tried to answer these questions, by going over many of the common triangle configurations in the olympiad geometry of today. In olympiad culture, there are a lot of colloquialisms, such as all of the names above, and I have tried to use these in the handout. However, you obviously can't cite "Ex Point" as well known, and in that a secondary purpose of the handout is to act like a list of some of what I think are the best proofs of so called "well known" facts, for when they're necessary on an olympiad. I have included 100 problems in roughly increasing order of difficulty at the end, 19 of these having solutions. I hope that these solutions, and the hints that accompany 36 of the problems, show what motivation there might be to the solution of a triangle-configurational geometry problem, once one knows all of the facts about a such configuration.
I would like to give thanks to amar_04, who proofread it without any incentive. amar_04 is really pro and has seen a lot of nice problems, and without their suggestions this handout wouldn't be what it is. I would also like to give thanks to The_Turtle for letting me use this quote, to show what I'd like to try to help solve.
The_Turtle wrote:
Quote:
everyone knows all of the configs
Without further ado, here is the link. If you're wondering what the title is all about, the title is just meant to be slightly funny, nothing more. If you see any mistakes, you can put them over here, I'd like to keep all of them organized.
Thanks,
i3435
