[/quote]
,
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 
and 
is acute and let 
be its orthocenter. The points 
and 
on the lines 
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 
with 
so that 
The point 
satisfies 
the point 
satisfies 
and the point 
on the line 
Let 
and 
be the circumcenters of the triangles 
and 
respectively. Prove that the lines 
and 
are concurrent.
,
as the product of the digits of 
.
be a function satisfying the following conditions:
,
and 
,
;
is surjective.
.
and 
.
and 
respectively. The point 
such that 
is equilateral. Let 
and 
.
.
Y by
Let 
be a natural number. At a mathematical olympiad training camp the same 
courses are organised every day. Each student takes exactly one of the courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of , the largest possible number of students at the camp?
be a natural number. At a mathematical olympiad training camp the same 
courses are organised every day. Each student takes exactly one of the courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of , the largest possible number of students at the camp?
for the courses and everyone takes the final course together, and for a bound note that if we cyclically shift a permutation of courses it doesn't have an intersection with itself, so out of every cycle we have at most 1 permutation and there are 
.
on the first day. Then, for days 
,
,
.
students, and we're done.
