[/quote]
,
denote the set of positive integers
such that for all 
it holds that 
satisfying 
holds for all real numbers 
.
be positive real numbers such that 
Prove that:
and 
be two sequences defined by:![\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]](http://latex.artofproblemsolving.com/a/a/0/aa0a5c5acedc990eb345fb50a8395db05307a4e7.png)
for all 
,
and 
.![\[
a_{2024} < 5.
\]](http://latex.artofproblemsolving.com/f/c/a/fca6e6630f33429b5a7ea867d639a2e29ff881f2.png)
and write the numbers 
down on a board. A move consists in erasing any two numbers 
and 
,
and 
on the board, and then removing repetitions (e.g., if the board contained the numbers 
,
and 
,
)
with real coefficients of degree 
)![\[
Q(x) \;=\; P(x) \;+\; a\,P'(x) \;+\; a^2\,P''(x) \;+\; \dots \;+\; a^n\,P^{(n)}(x)
\]](http://latex.artofproblemsolving.com/6/5/e/65e5e4eefcadd5d0019b4929b3472cdf203c13aa.png)
has no real roots.
and 
,
,
the pair 
if for any elements 
we have 
.
be the greatest possible number of elements of a set that escapes the pair 
.
with incircle 
,
be the incenter and 
be the point where 
.
be the point on 
with 
and 
be the orthocenter of 
,
on 
also satisfies 
.
,
,
satisfying 
the following inequality holds:
always exists but not necessarily zero.
Y by
Let 
is differentiable such that 
Then which of the following is correct?
is differentiable such that 
Then which of the following is correct?
always exists but not necessarily zero.- always exists and is equal to zero.
- may not exist.
- exists if
is twice differentiable.
This post has been edited 1 time. Last edited by Mathlover08092002, Apr 10, 2020, 7:51 PM
in the following way:![\begin{align*}
I_1 &= \Big[0, 2\pi\Big] \text{of length}\ l = 2\pi \\
I_2 &= \Big[2\pi, 3\pi\Big] \text{of length} \ \frac{l}{2} = \pi\\
I_3 &= \Big[3\pi, 3\pi + \frac{2\pi}{3}\Big] \ \text{of lenght} \ \frac{l}{3} \ \text{and so on}
\end{align*}](http://latex.artofproblemsolving.com/6/1/5/61552a4cc1991213b999d611be64613e3f3bd746.png)
Define the following function:
where 
for ![$x \in \Big[0, 2\pi\Big] $](http://latex.artofproblemsolving.com/9/f/c/9fcf60fc114ff321ea0c6d5a68c0abc4ffa40a46.png)
and
for 
.
has the derivative equal to 
in every point 
and equal to 
in every point 
,
.
oscillates between 
,
