[/quote]
,
.![\[
(2^x - 1)(5^x - 1) = y^n
\]](http://latex.artofproblemsolving.com/f/8/f/f8f013a96863bc5f39aa23edb98eed9aec54a17f.png)
have no positive integer solution 
.
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x^2 + y) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
such that:
be a positive integer, and let 
denote the integers modulo 
satisfying 
for all 
.
such that 
divides 
for any integers 
and there exists a polynomial 
with integer coefficients, such that 
for all 
.
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
for which: Every 
.
such that 
then find minimal value of 
has circumcircle 
and incircle 
,
at 
,
is an arbitrary point on 
meets 
at 
,
meets 
meets 
.
is a point on 
are colinear. Let 
be the intersection of 
and 
is tangent to 
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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
