[/quote]
,
for all 
.
.
the point 
is located on the side ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
,
is the foot of the perpendicular from 
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.
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:
of the triangle 
.
the orthogonal projection of 
.
and 
are such that 
is a paralellogram, find the locus of point 
.
Y by
A, B, C, and O are four points in the plane such that
and
.
Let
be a point on 
, and let 
be a line passing through such that
and
.
The line intersects 
at 
and the circumcircle of triangle 
at 
(
).
Show that the circumcircles of triangles
and 
are tangent at .
and
.Let
be a point on 
, and let 
be a line passing through such that
and
.The line
intersects 
at 
and the circumcircle of triangle 
at 
(
).Show that the circumcircles of triangles
and 
are tangent at .
