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be three non-negative real numbers satisfying 
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 
.
, such that: 
For all 
.
is 
, maybe even with rather elementary means? As usual, 
denotes the 
-th prime number. The problem of that limit came up in my partial solution of this problem: https://artofproblemsolving.com/community/c7h3495516.
. Define 
as the average of all 
for 
. Determine the value of ![\[ a = \lim_{k \to \infty} \sup_{n \geq k} \frac{a_n}{n^4} .\]](http://latex.artofproblemsolving.com/5/2/d/52d93af7aafa6657689246d9a74c8eb05d96efa3.png)
be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could 
be a prime number?
be an open interval and 
a twice differentiable function such that 
for any 
Prove that 
for any
which verify relation![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
has property 
if for any real number 
there exists a 
such that 
for all 
that is not monotone on 
is nondecreasing.
is field. Define addition of elements of 
coordinate wise and for 
and 
define 
. is a vector space over field 
points?
points?
, with their second derivative being continuous, such that the following holds for all : ![\[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]](http://latex.artofproblemsolving.com/0/2/9/029a935e28ca0a9b36e65702689cca91eeb8f614.png)
the bisectors of angles 
and 
meet the opposite sides at 
and 
respectively. Show that if the angle at 
is greater than 
, then 
.
,
,
,
.
.
