[/quote]
,
and 
Prove that
.
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 
can be inscribed within a circle.
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 
.
and the circle 
of radius 
Let 
The line 
intersects 
Let 
Let 
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)
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
Given a triangle 
inscribed in circle 
(with center 
and radius 
) with 
and let 
be a diameter of the circle 
. The perpendicular bisector of intersects line 
at point 
and line 
at point 
. Line 
intersects the circle at point 
. Let 
be the second intersection point of cicumcircles 
of triangle 
, and 
of triangle 
. Prove that lines 
and 
pass through the same point.
inscribed in circle 
(with center 
and radius 
) with 
and let 
be a diameter of the circle 
. The perpendicular bisector of intersects line 
at point 
and line 
at point 
. Line 
intersects the circle at point 
. Let 
be the second intersection point of cicumcircles 
of triangle 
, and 
of triangle 
. Prove that lines 
and 
pass through the same point.This post has been edited 1 time. Last edited by parmenides51, Nov 18, 2020, 4:45 PM
Reason: typo: ... and line AB at point N
Reason: typo: ... and line AB at point N
gives that 
is cyclic.
gives that 
is cyclic. And
so 
is also cyclic and by Radical Axis Theorem in 
