[/quote]
,
the points 
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
such that 
.
such that 
.
.
be a cyclic quadrilateral with circumcenter 
,
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
,
lies between 
and 
lies between 
is acute and let 
be its orthocenter. The points 
and 
and 
.
,
.
which satisfy the condition![\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]](http://latex.artofproblemsolving.com/4/b/d/4bddce124cf0da82cb8bb6c6ae2325239354f27c.png)
Find all possible values of 
be a positive integer and 
be a positive integer coprime to 
.
,
,
Find, in terms of 
,
for which there exists an index 
such that 
is divisible by 
.
table are initially white. Alice and Bob play a game. First Alice paints 
rows and 
Y by cursed_tangent1434, NicoN9, EeEeRUT
I hope posting this in Site Support is fine? I'm not sure whether an HSO admin is doing this but-
The EGMO 2025 problem 1 and problem 2 were originally posted by me on the day of the contest and found deleted (not by me) yesterday. Someone else noticed this and posted the problems again, but these threads were deleted once more.
The EGMO 2025 problem 1 and problem 2 were originally posted by me on the day of the contest and found deleted (not by me) yesterday. Someone else noticed this and posted the problems again, but these threads were deleted once more.
