[/quote]
,
Prove that
Let 
Prove that
and 
Prove that 
![$A \in \mathbb Q[x]$](http://latex.artofproblemsolving.com/d/1/5/d158f344cf30f8beb5f2199da53272bd8e304704.png)
with ![$\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$](http://latex.artofproblemsolving.com/1/1/9/119a7b617e09ba16bb266b44ab3c85acc529f88f.png)
and 
.
,
and positive real number 
,
,
,
,
holds.
be a acute triangle with 
.
are the orthocenter and excenter. Let 
be a point on the same side of 
as 
is equilateral. Let 
be a point on the same side of 
as 
is equilateral. Let 
be a point on the same side of 
as 
is equilateral. Let 
be the midpoint of 
.
are collinear.
be real numbers satisfying 
and 
Prove that 
Let 
and 
Prove that 
be reflections of 
across the line 
be the midpoints of 
,
be the reflections of 
and 
are natural numbers such that 
is divisible by 
and 
is divisible by 
,
.
so that 
The point 
the point 
and the point 
Let 
and 
be the circumcenters of the triangles 
and 
respectively. Prove that the lines 
and 
are concurrent.
be a positive integer and let 
,
,
be positive integers from set 
such that every number from this set occurs exactly once. Is it possible that numbers 
all have different remainders upon division by 
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Y by Adventure10, Mango247
Distinct points 
lie on an arbitrary line 
. 
is a point not lying on . A line passing through 
and parallel to 
intersects with 
in 
and a line passing through 
and parallel to 
intersects with 
in 
. Let 
be the intersection point of the circumcircles of 
and 
(
). Prove that 
.
lie on an arbitrary line 
. 
is a point not lying on . A line passing through 
and parallel to 
intersects with 
in 
and a line passing through 
and parallel to 
intersects with 
in 
. Let 
be the intersection point of the circumcircles of 
and 
(
). Prove that 
.
denote the infinite points of 
Parallel from 
cuts 
at 
By Desargues involution theorem, the opposite sidelines of the complete quadrangle 
form an involution on the line at infinity 
of 
and 
as desired.
and 
.
and we have
. So we have 
.
By noting that 
,
,
and by using sine rule, we get the desired result. 
and 
,
So, combing this with 
,
.
which implies 
, 
and 
, 
so we need to prove it 
we have: 
and 
.
at 
.
is cyclic. Now we have 
so we need to prove 
.
.
