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.
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.
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.
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 
with 
and the circle 
of radius 
centered at 
Let be the midpoint of 
The line 
intersects a second time at 
Let be a point on such that 
Let 
be the intersection of 
and Prove that 
, 
, 
be non-negative real numbers, no two of which are zero. Prove that :
with 
. Let 
points on sides 
, respectively, such that 
are angle bisectors on . is isosceles if and only if 
is right-angled isosceles.
and w.r.t center 
of the circle, be 
and 
respectively. then by simple angle chasing , we can see that 
forms a parallelogram . since diagonals of a parallelogram bisect each other 
midpoints of 
and 
concur at one point. (say at X)
thus the ratio turns out to be equal to
from the quadrilateral and an inscribed circle, centre .
; then![\[r= 2 \cdot \sqrt{\frac{(M-abc)(M-bcd)(M-cda)(M-dab)}{abcd (ab+cd)(ac+bd)(ad+bc)}}\]](http://latex.artofproblemsolving.com/1/6/0/160b9c73f3e12ea7933112084d5bd75b95fa9868.png)
?
![\[AO = 5, BO =6, CO = 7, DO = 8.\]](http://latex.artofproblemsolving.com/6/b/5/6b5cece87816d6ca54c50ccd5a3128269a0bbbd8.png)
,
.
