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 
to 
and from 
to 
and 
as the reflections of 
be the circumcircle of 
.
the second intersection of line 
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 Adventure10
In a certain triangle 
, there are points 
and 
on sides 
and 
, respectively, such that if 
is the intersection of 
and 
, then both 
and 
are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle .
, there are points 
and 
on sides 
and 
, respectively, such that if 
is the intersection of 
and 
, then both 
and 
are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle .
be the intersection point of 
and 
and 
are the 
as inscribed angles. Hence 
is cyclic; similarly we prove that 
also is cyclic. We claim that 
and 
,
;
,
.
,
,
is an altitude. Similarly we prove that 
and 
are also altitudes.
as inscribed angles. Moreover, 
,
.
,
.
angles 
)
.
,
,
,
,
are the midpoints of 
,
,
,
and 
are congruent, so 
,
and 
we have 
,
,
,
,
.
cyclic means 
means 
,
as external angle in 
,
becomes 
,
