[/quote]
,
such that 
divides 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
and 
.
,
is a subset of 
such that every element of 
.
![\[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \]](http://latex.artofproblemsolving.com/0/9/d/09d80fe5c25c5c6ea57c545b68c7e11b319cbada.png)
with integer coefficients 
,
,
,
and 
such that ![\[ f(a)=f(b)=f(c)=f(d)=5, \]](http://latex.artofproblemsolving.com/8/2/d/82d0c5d0c7a6dd48802ad8fbe04b211278c1fa58.png)
show that there is no integer 
such that 
.
such that 
for all 
?
.
such that for all any 
denote the number of positive integers less than 
has at least 
solutions in 
such that for all any 
:
of functions from the set of positive integers to itself that satisfy ![\[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\]](http://latex.artofproblemsolving.com/5/4/8/5489ee420a448756636625801f2bc7dcce6f0ef8.png)
for every positive integer 
means 
.
be a function from the set of integers to the set of positive integers. Suppose that, for any two integers 
is divisible by 
.
,
is divisible by 
.
such that ![\[f(x+f(x)+2f(y))+f(2f(x)-y)=4x+f(y)\]](http://latex.artofproblemsolving.com/4/4/2/4423651bb829b92debd65a23f6cee3d4d4b8530f.png)
holds for all reals 
and 
.
in which each term is either 0 or 1 is called a binary sequence of length 
be the number of binary sequences of length 
be the number of binary sequences of length 
for all positive integers 
![\[ f\left( f(x) y + \frac xy \right) = xyf(x^2+y^2). \]](http://latex.artofproblemsolving.com/b/6/5/b656324b6499f7457b7b9eccbf5eaa01b6ad6e59.png)
,
and 
.
satisfies the following property:![\[
f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\}
\]](http://latex.artofproblemsolving.com/3/b/d/3bd9fbf82b0b1d98a3c68745bb3f159cd9e2529a.png)
and for all 
.
Y by doxuanlong15052000, Adventure10
Given triangle 
and let 
be the foot of angle bisectors of 
,respectively.
lie on 
such that 
. Let 
be the foot of 
-altitude on 
.
Points
lie on such that triangles 
are correspondingly similiar (with the given order of vertices) such that 
and 
.
Show that:
and let 
be the foot of angle bisectors of 
,respectively.
lie on 
such that 
. Let 
be the foot of 
-altitude on 
.Points
lie on such that triangles 
are correspondingly similiar (with the given order of vertices) such that 
and 
.Show that:
be the foot of angle bisectors of 
let 
the proyections of 
at 
a point at 
such that 
is tangent to 
,
is cyclical 
and 
and 
points at 
and 
and 
is evident that 
foot of 
with 
and 
intersection of 
with 
the point on 
and 
reflection of 
about 
intersection of 
we can see that 
so 
are cyclic .
and we're done 
