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.
such that for all 
we have:
be an acute triangle inscribed in its circumcircle 
, and let 
be its incircle. Let 
be the point where the 
incircle of triangle 
touches 
. Suppose line 
intersects segment 
at 
, and intersects line 
at 
. Let the line through 
perpendicular to intersect line 
at 
. Prove that: ![\[AI \parallel PA'.\]](http://latex.artofproblemsolving.com/0/2/6/0264e64b5edff4aca1e35037c5b0f86763c3b4e7.png)
in its exterior
we draw the triangles 
so that 
, 
, 
.
and
is an arithmetic sequence of positive integers, and 
is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
such that the following statement holds: for every integer 
and sequence of non-integer positive real numbers 
satisfying 
it's possible to choose positive integers 
such that
, either 
or 
, and
(Here 
denotes the floor function, as usual.)
such that the following holds: if we have any 
such coins with total value 
, then we can always split them into two stacks of coins each such that the absolute difference between the total values of the two stacks is at most 
.
be positive real numbers such that 
. Prove that![\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]](http://latex.artofproblemsolving.com/6/5/5/65558feba82e32266d3d3cbdfea85e079483403f.png)
be an acute angled triangle with orthocenter 
and 
. Let 
be any other point on the arc 
of the circumcircle of 
and let line 
intersect line 
at 
and let line 
intersect line 
at 
. Let the circumcircles of 
and intersect again at 
(
). Let the lines 
intersect the circumcircle of 
again at 
(
). Prove that the lines 
concur.
be the circumcircle of the triangle 
. 
be the antipode of 
in . Angle bisector of angle 
meets 
and 
at 
and 
. Let 
be the midpoind of the arc 
. 
. 
and 
be the Thebault circles of the cevian 
. Prove that circumcircle of the triangle 
tangent to , and . with incircle 
tangent to 
at points 
, respectively. Let 
be a point such that its isogonal conjugate lies on the line 
(where 
is the circumcenter and 
the incenter of ). The line 
intersects segments 
and 
at points 
and 
, respectively, such that the circle with diameter 
meets at points 
and 
.
is tangent to the incircle at some point .
corresponding to vertices 
. Prove that the lines 
are concurrent.
of an acute-angled triangle meet the internal bisector of angle 
at points 
, and the external bisector of this angle at points 
. Prove that the orthocenters of triangles 
and the vertex are collinear.
of an acute-angled triangle meet the internal bisector of angle at points , and the external bisector of this angle at points . Prove that the orthocenters of triangles and the vertex are collinear.
, 
.Let the internal,external bisector of angle meet 
at 
resp.Let 
be the orthocenters of 
resp.
is a rectangle with circumcircle 
and 
are collinear in this order
and 
is collinear
orthocenters formed by any four lines in the plane (in general position) are collinear. Take the four lines to be the two altitudes and two angle bisectors. is the orthocenter of two of the triangles (as two bisectors are perpendicular). So we are done..
then prove that 
