[/quote]
,
are lengths of sides of some triangle. Prove the inequality
where 
.
is the circumcenter of triangle 
is the projection of point 
onto line 
.
,
are 
,
,
,
intersects 
and 
at points 
and 
,
.
-
,
,
,
according to the following rule: If 
,
,
if 
(taking 
)
is the symbol other than 
if 
.
for which there exists some integer 
such that 
.
be an acute triangle with orthocenter 
.
,
and 
be the feet of the altitudes of triangle 
,
,
respectively. Let 
and 
be the midpoints of 
and 
,
be the intersection of lines 
and 
.
.
and 
Prove that
and 
be two prime numbers, with 
.
such that the polynomial 
is divisible by the polynomial 
with 
is a multiple of 
.
be the circumcircle of acute triangle 
and 
are on segments 
and 
respectively such that 
.
and 
intersect minor arcs 
and 
respectively. Prove that lines 
and 
are either parallel or they are the same line.
and circumcircle 
.
is the midpoint of small arc 
o
is the projection of 
with center 
is internally tangent to 
meets 
,
bisects 
.
,
.
at 
.
,
meet 
.
at 
.
is tangent to 
be two nonzero real numbers such that 
Consider the functions 
and 
satisfying the condition ![$$g(f(x+by))=a[f(x)+2yg(x)]+2xg(y)+bg(y)[y+g(y)],\quad\forall x,y\in\mathbb{R}.$$](http://latex.artofproblemsolving.com/3/c/9/3c974cb76aae7f529dcd85202b0c7d441a07d002.png)
(a) Prove that there exists a real number 
(b) Find all functions 
and 
that satisfy the given condition.
, such that 
Prove that :
equality cases : ?
be a square and a point 
is equilateral. Evaluate 
to 
(inclusive) are selected randomly without repetitions, then either all of them will be odd, or all will be divisible by 
,
with 
is coprime to 
.
,
satisfying 
Prove that 
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Hello everyone,
Over the past few months I have written an article developing much of the theory on Cyclotomic Polynomials and many applications to Olympiad math. Its been almost finished for the past few weeks, but schoolwork delayed me finishing it for a very long time but now it is finished! There are extremely interesting and applicable to a pretty reasonable set of Olympiad Number Theory problems.
Below is a link to a copy in dropbox, however for some reason the hyperlinks don't render correctly on it which is pretty annoying. Because of this, a copy of the article is also attached to this post which hopefully the hyperlinks work correctly on.
I have also attached the tex file for my article in case they do not render correctly on the pdf, because then you can compile the document yourself. It also is attached to serve as a LaTeX sort of guide, because I myself struggled for quite a while to learn sufficient LaTeX to make a pdf which looked nice.
If you find any errors/typos, please e-mail me or private message the errors to me so I can correct them. Posting them onto this thread is fine as well, but I would prefer you not do that because I want this thread to be used for discussion of the exercises.
I hope you enjoy this article!
Dropbox Version
EDIT : I have attached newer versions of the tex file/pdf now.
EDIT2: I have attached even newer versions.
Over the past few months I have written an article developing much of the theory on Cyclotomic Polynomials and many applications to Olympiad math. Its been almost finished for the past few weeks, but schoolwork delayed me finishing it for a very long time but now it is finished! There are extremely interesting and applicable to a pretty reasonable set of Olympiad Number Theory problems.
Below is a link to a copy in dropbox, however for some reason the hyperlinks don't render correctly on it which is pretty annoying. Because of this, a copy of the article is also attached to this post which hopefully the hyperlinks work correctly on.
I have also attached the tex file for my article in case they do not render correctly on the pdf, because then you can compile the document yourself. It also is attached to serve as a LaTeX sort of guide, because I myself struggled for quite a while to learn sufficient LaTeX to make a pdf which looked nice.
If you find any errors/typos, please e-mail me or private message the errors to me so I can correct them. Posting them onto this thread is fine as well, but I would prefer you not do that because I want this thread to be used for discussion of the exercises.
I hope you enjoy this article!
Dropbox Version
EDIT : I have attached newer versions of the tex file/pdf now.
EDIT2: I have attached even newer versions.
This post has been edited 4 times. Last edited by dinoboy, May 16, 2014, 8:49 PM
has degree 
I haven't heard about the irreducible polynomial of 
for 
odd primes after one proves it is true for 
,
over 
.
.![\[\prod_{d|3}\Phi_{3d}(2) = \Phi_3(2)\Phi_9(2) = (2^2 + 2 + 1)(2^6 + 2^3 + 1) = 7\cdot 73 \ne 73\]](http://latex.artofproblemsolving.com/a/2/3/a23595f6828f0e0625b08252134507a3aed5ff51.png)
,
