AoPS Academy
be a positive integer and let 
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of around the circle such that the product of any two neighbors is of the form 
for some positive integer 
.
+1 w
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a fixed triangle, and let 
, 
, 
be the midpoints of sides 
, 
, 
, respectively. Let 
be a variable point on the circumcircle. Let lines 
, 
, 
meet the circumcircle again at 
, 
, 
, respectively. Assume that the points 
, 
, 
, , , are distinct, and lines 
, 
, 
form a triangle. Prove that the area of this triangle does not depend on .
where 
is a natural number.What is the remainder when is divided by 
The altitudes 
, intersect
. The tangents at 
and 
of intersect at . Let 
be the midpoint of 
. 
intersects 
, 
intersects 
at 
lie on the same circle.
intersects the circle at the second point 
The line intersects at the second point 
. Prove that 
are collinear.
is . Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that 
.
?
Prove that
,
is
Prove that
Where 
Subject to the constraints:
be the collection thus formed, and let 
be the collection formed by taking each circle in 
times the radius with the same center.
times the area of 
the area of