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 a triangle with 
and the rea is 
Find the minimum of 
, 
,..., 
.
. Find the minimum value of 
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 the feet of the altitudes of the triangle from 
, respectively. 
is the point on the circumcircle of the triangle , 
is the orthocenter of the triangle , and the incircle of triangle 
has radius 
. Let 
be the point such that and are on the opposite side of 
, line 
is perpendicular to the line , and the distance from to line is . Define 
and 
similarly. Prove that 
are collinear.
be real numbers such that 
and 
. Find the largest possible value of 
.
be a positive integer. In an 
table, an upright path is a sequence of adjacent cells starting from the southwest corner to the northeast corner such that the next cell is either on the top or on the right of the previous cell. Find the smallest number of grids one needs to color in an table such that there exists only one possible upright path not containing any colored cells.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let be its orthocentre. Points and 
lie on lines and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
, 
, 
, 
of positive integers such that for every central sequence 
, 
, 
, , there are infinitely many positive integers with 
.
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
be a triangle, and let 
and be the orthogonal projections of 
onto the internal bisectors from 
and . Prove that is parallel to 
.
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 non-isosceles triangle with orthocenter , circumcenter 
. Let 
be midpoints of 
and 
be reflections of across 
. Prove that triangle 
have the same centroid.
