grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
.
and 
be the feet of the altitudes from 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
?
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
Y by
Two circles intersect at points 
and 
. On the first circle, an arbitrary point 
is selected, different from and and lying inside the second circle, the rays 
and 
re-intersect the second circle at points 
and 
, respectively. Prove that the line passing through and the center of the first circle is perpendicular to the 
.
and 
. On the first circle, an arbitrary point 
is selected, different from and and lying inside the second circle, the rays 
and 
re-intersect the second circle at points 
and 
, respectively. Prove that the line passing through and the center of the first circle is perpendicular to the 
.This post has been edited 1 time. Last edited by parmenides51, Jul 4, 2020, 1:37 PM
Reason: title typo
Reason: title typo
,
is the center of the first circle.
(in other words; 
)
and 
.
and 
.
,
.
,![\[
\angle{BAE}+\angle{ABE}=90^\circ.
\]](http://latex.artofproblemsolving.com/3/5/c/35c5c74c3e0d506dcab38468a529fe39a8b4642e.png)
