AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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)
+1 w
be a point interior to triangle 
(with 
). The lines 
, 
and 
meet again its circumcircle 
at 
, 
, respectively 
. The tangent line at 
to meets the line 
at 
. Show that from 
follows 
.
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of .
of positive integers such that 
is an integer.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist., its circumcircle, and its incenter are drawn on the plane. Construct the circumcenter of using only a ruler.
such that:
is strictly increasing,
such that 
for all 
,
, there exists 
such that![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](http://latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
has two isogonal conjugate points and 
. The circle 
intersects circle 
at 
, and the circle 
intersects circle 
at 
. Prove that 
and are isogonal conjugates in triangle . is the circle with diameter , Circle is the circle with diameter 
.
be a real or complex inner product space. Suppose that 
and that 
Then
, s.t :
and 
be the midpoint of 
and 
the vertices of 
go to the respective midpoints
be the center of 
then 
and 
Prove that
satisfying
Prove
satisfying
Prove
.
be the touchpoints of the incircle with 
,
,
is the Euler line of 
.
the unit circle. Then 
.
the orthocenter of 
,
.
are collinear 
.
,