.
Let 
.
Let 
.
Let 
.
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
be a positive integer. We say that a positive integer 
is 
is divisible by 
for all positive integers 
with 
.
is not 
is prime.
satisfy 
for all 
.
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
is a finite group of 
matrices with respect to multiplication. Prove that:
that 
be an infinite set of translated copies (i.e., obtained by parallel translation) of a given ellipse 
in the plane, and let 
be a fixed straight line. It is known that every straight line parallel to 
fibonacci numbers 
)
and 
that 
'strong' and 
'strong'
and the distribution function is 
then
and find the expression of 
be a continuous function . Suppose![\[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\]](http://latex.artofproblemsolving.com/c/0/c/c0c48c4259d4e98562c21a23cf4a6841704b8e82.png)
and all 
. Then show that there exists a constant 
such that 
, 
,
Y by Maths_Guy, Adventure10
Let 
be the incircle of a triangle 
. The line passing though the incenter 
and parallel to 
meets at 
and 
( lies in the same semi plane with respect to 
as 
). The lines 
and 
meet at 
. The points 
and 
are defined similarly. prove that 
concur.
be the incircle of a triangle 
. The line passing though the incenter 
and parallel to 
meets at 
and 
( lies in the same semi plane with respect to 
as 
). The lines 
and 
meet at 
. The points 
and 
are defined similarly. prove that 
concur.
are isogonally conjugates. The same result is valid for the pair of lines 
and 
.
and 
.
and 
hence the desired ratio product equals 
.
