[/quote]
,
,
be positive integers such that 
is a multiple of 
for all positive integers 
.
.
be a positive integer and 
be a family of at most 
with the following property: we can consider 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
such that the equation 
has an integer solution 
.
such tha
(Adit Aggarwal, India.)
such that 
and 
for all reals 
.
and some point 
on 
.
intersects the circumcircle of 
at 
and the perpendicular from 
intersects the circumcircle of 
at 
.
does not depend on the choice of 
of parallelogram 
a point 
is chosen. The perpendicular from 
to 
and the perpendicular from 
to 
intersect at 
is the midpoint of 
.
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
be a convex hexagon, such that the triangles 
and 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
is prime.
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
is an integer.
?
such that![\[ f(x)f\left(x + 4f(y)\right) = xf\left(x + 3y\right) + f(x)f(y) \]](http://latex.artofproblemsolving.com/d/b/1/db14b64aa71cb3c8dbf53707f719f5d9a43bd7e4.png)
for any positive real numbers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](http://latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers 
with 
Y by Adventure10, Mango247
Let 
be one of the intersection points of two circles in the plane. In each of the circles a diameter is drawn parallel to the tangent to the other circle at . Prove that the endpoints of the diameters lie on a circle.
(S. Berlov)
be one of the intersection points of two circles in the plane. In each of the circles a diameter is drawn parallel to the tangent to the other circle at . Prove that the endpoints of the diameters lie on a circle.(S. Berlov)
