Theorems/Formula/Factor Tricks

[zefuri]

List of Theorems/Formulae:

ill list a couple

Ceva(look it up cuz it only works for a diagram)
Menelaus(same)
Stewarts

[stupidityismygam]

Factorizations: Here are some cool ones

Sophie Germain: [unparseable or potentially dangerous latex formula][/tex]

Clever: [unparseable or potentially dangerous latex formula]
or
[unparseable or potentially dangerous latex formula]

[zefuri]

Dirichlet's if a and b are relatively prime then a,a+b,a+2b...will have an infinite number of primes.

AM-GM

[unparseable or potentially dangerous latex formula] (i think theres one more but i forgot)

[stupidityismygam]

Geometry concepts:

Excircles, incircles
Miguel Points, Brochard Points
Spiral Similarity
Angle Chasing
Centriod, Orthocenter, Incenter, Circumcenter
Cyclic Quads

Number Theory:
Wilson's Theorem
Fermat's Little Theorem
Fermat's Last Theorem
Chinese Remainder Theorem
Order, Generators
Diophantine Equations

Algebra:
Recursions
Generatingfunctionology
Polynomials
Factorizations

Trig and Others:
All half angle, double angle, product to sum, sum identities
DeMoivre's
Roots of Unity
Inequalities
Ball and Bins Argument
Conditional Probability
Pigeonhole
Graph Theory/Set Theory

I'll post information about this later, but this is just a brief overview of what you should be looking for
Suggested Study List:
ACoPS
101 Algebra
102 Combinatorics
103 Trigonometry
104 Number Theory
Olympiad Books
AoPS 2 (has general info)
AoPS Mildorf inequalities Link:

[stupidityismygam]

Wilson's Theorem:

Let [unparseable or potentially dangerous latex formula] be an integer greater than one. p is prime iff [unparseable or potentially dangerous latex formula]

Fermats Last Theorem:

There are no non-zero integer solutions (x,y,z) to [unparseable or potentially dangerous latex formula] when
[unparseable or potentially dangerous latex formula]

De Moivre's:

[unparseable or potentially dangerous latex formula]

Balls and Bins Argument:

For n balls, and k bins, the total arrangements are
[unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula]

I'll give an example for Balls and Bins...

Find the total number of solutions to x+y+z=20 . given that x,y,z are integers.

There are 3 bins, and 20 balls (see if you can see why) So it is just [unparseable or potentially dangerous latex formula]

[makashi]

It may also pay to familiarize with some general problem solving strategies, such as induction, even though those are usually the ones that are more intuitive. From what I've heard, the CAMC are big fans of, to borrow a phrase from Arnold Ross, thinking deeply of simple things. Consequently, there are problems which can be solved without usually high powered theorems. For example, the first problem last year was simply Pigeonhole principle, but the difficulty was finding the right pigeons and holes. The second problem was monovariance (or extreme principle, whichever you prefer) disguised as an optimization problem. Neither of the solutions cited any theorems. It is my impression that the cutoff for red MOP is related to the number of these problems, as it is usually the youngest, and therefore likely lesser experienced, group. That may be why last year qualifying for red group required solving more than one problem for the first time in a while.

[zefuri]

but hasnt htere only been 2 years of red mop so far?
i know induction and stuff, havent used pigeonhole that much but im familiar with it, dont know what monovariance/extreme principle is...

[bradp]

How is knowing these little factoring tricks any better than knowing what a happy number is?

[stupidityismygam]

because they are actually useful for something...

if you can write me an AIME/USAMO problem that revelant to the solution is happy numbers...then i will give you a cookie

[stupidityismygam]

NOTE: without defining it in the beginning...

such as A happy number is blahblahblah, find with proof the largest 3 digit happy number

[zefuri]

*agrees with stephen*

[bradp]

I probably can't write a USAMO problem... my knowledge of Simon's Favorite Factoring Trick isn't extensive enough.

[stupidityismygam]

[zefuri]

actually it takes a lot more skills than just math to be able to write problems because you ahve to make it not crazy and like on the correct difficulty level which is harder than you think...

[bradp]

I have some NS tests I wrote if anyone wants to take a gander.

[stupidityismygam]

[zefuri]

Why do you mention Simon's Favorite Factoring Trick so much(just curious) and uhh its not even "respected" because it started on aops...isnt that the one like
[unparseable or potentially dangerous latex formula]
or am i thinking of something else 0.o

[stupidityismygam]

zefuri you are correct

[bradp]

I just think it's funny

I mean first all.. Simon is funny
And that fact that it is his favorite is also funny
i'll leave it alone

[zefuri]

But Simon is also pr0

[AuSmith]

[zefuri]

well like i am pretty sure they dont consider it necessary in competitions like USAMO and IMO and its quite trivial anyway

[AuSmith]

[zefuri]

well i meant in terms of citing...you can just say you factor instead of like saying you used SFFT

[AuSmith]