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.
, let 
be the 
-excenter. Points 
and 
are placed on line 
such that 
is between and 
, and is between and . Moreover, and are the contact points of with the -excircle of triangles 
and 
, respectively. Let 
be the -excenter of triangle 
, and let 
be the reflection of the orthocenter of triangle with respect to its circumcenter. Prove that , , and are collinear.
is sabroso, since 
. Find all two-digit sabroso numbers.
satisfy the following two conditions:
exist such that 
.
, exist such that 
.
then prove 
Equality holds iff 
or 
and 
.
satisfies the following three conditions:
All elements of are a subset of 
.
.
is the foot of the altitude from 
of triangle 
. On the lines 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.
.
be non-negative real numbers such that 
Prove that
, sequence 
satisfies the equation:![$\sum_{k=1}^{n}\frac{1}{2}(1-(-1)^{[\frac{n}{k}]})a_{k}=1$](http://latex.artofproblemsolving.com/3/2/1/3216129b3e42dfb49eb8bb764fafa1b9fbf28882.png)
, find the value of 
. such that there exist two permutations 
and 
of the set 
, satisfying the condition
for all 
. with 
. Let 
be a point of 
such that 
is tangent to the circumcircle of . Let 
and 
be the circumcenters of triangles 
and 
, respectively, and let 
be the midpoints 
. Prove that the line tangent to the circumcircle of 
through is also tangent to the circumcircle of .
such that for all 
, we have 
.
be a function over reals sich that 
is surjective.
for all reals 
and 
Prove that
.