AoPS Academy
be a positive integer, and consider the equation:
Answer the following questions. You may assume that 
is known.
Determine the number of real solutions of equation 
for each .
For 
, let 
be the segond largest real solution of .
Find 
such that 
Find 
, where 
is defined as
be reals with 
and 
. Prove that
![$a_n=\sqrt[n]{f(n)} , n \geq 2$](http://latex.artofproblemsolving.com/6/0/a/60aaa22c36ebc7f68b1dfaa3db07c6f3046c7cf0.png)
where 
is a bijection from 
to itself.I need to find the limit of this sequence when tends to infinity , in the solution my friend gave to me he said that it’s trivial that the limit is greater or equal than 1 can someone tell me why?
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
people seated around a circular table, and 
cookies
be one of these people. Find the least such that no matter cookies are distributed initially, there is a strategy to pass receives at least one cookie.
be non-negative reals such that 
.
and 
.
.
id defined by![\[
f(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
\]](http://latex.artofproblemsolving.com/0/2/c/02c7b4af3cac77da8e3cc102c79a7db2e95f437f.png)
now what is CDF or cumulative distribution function ![$F_X(x)=P[X\leq x]$](http://latex.artofproblemsolving.com/7/9/1/79150c7527581feec873c9c790f8aeddc8f8efb8.png)
.
,
.
.
where 
is the CDF of the standard normal distribution.