[/quote]
,
and 
Prove that
such that 
and 
for all reals 
.
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 
.
,
be positive integers such that 
is a multiple of 
for all positive integers 
.
.
be a positive integer and 
be a family of at most 
with the following property: we can consider 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
such that the equation 
has an integer solution 
.
such tha
(Adit Aggarwal, India.)
and some point 
on 
.
intersects the circumcircle of 
at 
and the perpendicular from 
intersects the circumcircle of 
at 
.
does not depend on the choice of 
of parallelogram 
a point 
is chosen. The perpendicular from 
to 
and the perpendicular from 
to 
intersect at 
is the midpoint of 
.
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 
be a convex hexagon, such that the triangles 
and 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
is prime.
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
is an integer.
?
Y by AlastorMoody, Adventure10
grade IX P4, X P3
The bisectors of the angles
and 
of the triangle 
intersect the circumscirbed circle of this triangle at the points 
and 
, respectively. The straight line passing through the center of the inscribed circle of triangle parallel to the side of 
, intersects with the line 
at 
. Prove that the line 
is tangent to the circumcircle of the triangle .
grade XI P4
The bisectors of the angles and of the triangle intersect the sides at the points 
and 
, and the circumcircle of this triangle at points and respectively. Straight lines 
and intersect at point . Prove that the segment connecting with the center inscribed circles of triangle , parallel to .
The bisectors of the angles
and 
of the triangle 
intersect the circumscirbed circle of this triangle at the points 
and 
, respectively. The straight line passing through the center of the inscribed circle of triangle parallel to the side of 
, intersects with the line 
at 
. Prove that the line 
is tangent to the circumcircle of the triangle .grade XI P4
The bisectors of the angles
and of the triangle intersect the sides at the points 
and 
, and the circumcircle of this triangle at points and respectively. Straight lines 
and intersect at point . Prove that the segment connecting with the center inscribed circles of triangle , parallel to .This post has been edited 3 times. Last edited by parmenides51, May 7, 2019, 4:02 PM
Reason: added 2nd similar problem
Reason: added 2nd similar problem
and 
intersect its circumcircle at 
.
at 
.
is tangent to circumcircle of 
be the circum-incentral triangle WRT 
![[asy]
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[/asy]](http://latex.artofproblemsolving.com/9/a/c/9ac345f4444272970eeffc4254f6c5d16562807e.png)
is cyclic. Let 
and 
