[/quote]
,
.
to 
and from 
to 
and 
as the reflections of 
be the circumcircle of 
.
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
Let 
be a circle, and let 
be a line segment such that 
lie on and 
is a point outside . Let 
be a point on such that 
is tangent to and let 
be the point on such that 
is a diameter of and intersects 
inside . Suppose that the lines 
and 
intersect at 
. Prove that 
is perpendicular to 
.
be a circle, and let 
be a line segment such that 
lie on and 
is a point outside . Let 
be a point on such that 
is tangent to and let 
be the point on such that 
is a diameter of and intersects 
inside . Suppose that the lines 
and 
intersect at 
. Prove that 
is perpendicular to 
.
the symmetrical points of 
,
.
.
is cyclic 
and 
.
,
is the radii of circle 
.
and 
.
passes on 
,
,
.
we get 
,
.
.
.
,
.
.
by controversial 
,
,
,
.
,
,![$ \frac{DE}{EN}=\frac{[DCE]}{[NCE]}=\frac{DC}{CN} \cdot {\frac{sin{\angle DCE}}{sin{\angle NCE}}=ctgy \cdot {tg{\angle{DCE}}}}$](http://latex.artofproblemsolving.com/9/7/2/972671d73a6906b744c176997b36510ed84a9675.png)
.
,
.
,
,
,
