[/quote]
,
.
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
and 
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.
and 
Prove that
have,where 
and 
are natural numbers.
be the set of the positive real numbers. Find all functions 
such that 
for all positive real numbers 
and 
.
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 
.
with 
and the circle 
of radius 
Let 
be the midpoint of 
The line 
intersects 
Let 
Let 
be the intersection of 
and 
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
,
.
such that: 
.
![\[
1 = d_1 < d_2 < \cdots < d_{k-1} < d_k = n
\]](http://latex.artofproblemsolving.com/a/d/2/ad2cbfe9e2110de2fdb6308d93f58880bd79afbc.png)
are the divisors of ![\[
d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1}
\]](http://latex.artofproblemsolving.com/d/2/3/d239210af0c2aa005ee1e5b820678d4aadc366c0.png)
forms a permutation of an arithmetic progression.(Kritesh Dhakal,Nepal)
be a finite set of integers. Prove that there exists a number 
depending on 
with integer coefficients the number of integers 
satisfying 
does not exceed 
.
such that 
and 
for all reals 
.
,
be positive integers such that 
is a multiple of 
for all positive integers 
.
Y by Adventure10, Mango247
Given a rectangle 
with a larger side 
. The circle centered at 
with radius intersects the line 
at points 
and 
. Prove that:
a) the circle circumscribed around the triangle
it tangent to a circle with a diameter of 
.
b) If
is the intersection point of these circles, then the points 
, , lie on one straight line.
with a larger side 
. The circle centered at 
with radius intersects the line 
at points 
and 
. Prove that:a) the circle circumscribed around the triangle
it tangent to a circle with a diameter of 
.b) If
is the intersection point of these circles, then the points 
, , lie on one straight line.
