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.
, 
be positive integers such that 
is a multiple of 
for all positive integers 
. Prove that 
.
be a positive integer and 
be a family of at most distinct subsets of the set 
with the following property: we can consider distinct points in the plane, labelled 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to subsets in for any 
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 with 
such that the equation 
has an integer solution 
.
such that
and 
(Adit Aggarwal, India.)
such that 
and 
for all reals 
.
and some point 
on 
. The perpendicular from to 
intersects the circumcircle of 
at 
and the perpendicular from to 
intersects the circumcircle of 
at 
. Show that the line 
does not depend on the choice of .(Shining Sun, USA)
of parallelogram 
a point 
is chosen. The perpendicular from 
to 
and the perpendicular from 
to 
intersect at . Point 
is the midpoint of 
. to passes through the center of parallelogram .
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 grid.
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid. that guarantees that Kritesh's job is possible?
be a convex hexagon, such that the triangles 
and 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
for which 
is prime.
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer the ratio 
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 .
be a positiv integer. Ana chooses the positive integers 
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 chosen by Ana, these can be determined by only knowing the sums 
with 
be a triangle and let be on such that 
. let 
meet circumcircle 
of triangle again at . prove that circumcentre of triangle 
lies on .
be the point which is intersection of 
and perpendicular bisector of 
. We can clearly get by 
and is angle bisector of 
that is perpendicular bisector of 
. Obviously we have circumcenter of triangle is the intersection of perpendicular bisectors of the sides. This completes solution. 
![[asy]
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[/asy]](http://latex.artofproblemsolving.com/5/e/7/5e75a9220c41b3ddb693d8e10d1790261ae281b6.png)
that passes through 
and intersect at a point 
. By a well known lemma (Fact 5), is the incenter of 
then by the same well known lemma (Fact 5) we are done.
cut at and at 
. We claim that is the reqd. circumcentre. is on the perpendicular bisector of , 
.
produced bisects 
. So, 
.
, proving our claim.
?
.
.
lies on