[/quote]
,
for all 
.
.
denote the set of positive integers
such that for all 
it holds that 
the point ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
,
on the line 
.
belongs to the line 
.
and 
are symmetric of the points 
with respect to the lines 
,
is square and the quadrilateral 
is rhombus.
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.
Y by
Let 
be a triangle with side lengths 
, 
, 
, incenter 
and circumcircle 
, and let 
be the circle that is tangent to 
, 
, at 
,
, 
, respectively. If 
,
, 
denote the incenters of 
, 
, 
, respectively, find the measure, in degrees, of the largest angle in 
.
be a triangle with side lengths 
, 
, 
, incenter 
and circumcircle 
, and let 
be the circle that is tangent to 
, 
, at 
,
, 
, respectively. If 
,
, 
denote the incenters of 
, 
, 
, respectively, find the measure, in degrees, of the largest angle in 
.
-
We can use some homothety centered at 
and 
is a right triangle. Therefore, the answer is 
.
