[/quote]
,
and 
Prove that
.
to 
to 
respectively.
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.
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 
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
Let 
be a triangle with 
and 
. Let 
be a point on the segment 
such that 
is perpendicular to . We mark the point 
on the segment 
so that 
. If segments and 
intersect at 
, prove that 
.
be a triangle with 
and 
. Let 
be a point on the segment 
such that 
is perpendicular to . We mark the point 
on the segment 
so that 
. If segments and 
intersect at 
, prove that 
.This post has been edited 1 time. Last edited by parmenides51, Apr 21, 2022, 1:55 PM
and 
.
intersects 
.
.
