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 an acute angled scalene triangle with circumcircle 
. Let 
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and , respectively, such that
Prove that 
and 
intersect on the circumcircle of 
.
+1 w
with circumcenter 
. Let be a point on 
such that 
. Denote 
the point on satisfying 
. If 
is the projection of onto 
, show that 
passes through the orthocenter of 
.
we consider the last digit different from zero in the decimal expansion of 
. The infinite sequence of these digits starts with 
. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
be circumscribed about circle 
, and let be the orthocenter of . The circle touches line at 
. The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point , and intersects the circumcircle of again at point . lies on the circumcircle of 
.
, 
, and 
. Points and of the sides and respectively, are such that 
and 
. Calculate the lenght of 
.
, then jumps in some order on each of the numbers 
exactly once and finally returns with his last jump to . Can the total distance he travelled with these 
jumps be a) 
, b) 
?
be any point on side of .Let 
and 
be points on 
and such that 
and 
respectively. Prove that the locus of circumcenter of 
is a line.
, such that 
Prove that :
equality cases : ?
board, show that there is a 
block on the board in which at least 3 of the squares are colored red.
intersect its circumcircle 
at 
. The tangets to from 
determine a triangle 
, where 
is relative to 
and so on. Prove that lines 
are concurrent at a point and that belongs to the euler line of .
intersect its circumcircle at . The tangets to from determine a triangle , where is relative to and so on. Prove that lines are concurrent at a point and that belongs to the euler line of .
be midpoints of 
, let 
be the centroid, and let be the circumcenter. It is easy to see 
are cyclic. Consider the circle , the circumcircle of 
, and the circumcircle of quadrilateral 
. The pairwise radical axes of these three circles are concurrent. This implies that the tangent to from , line 
, and line 
are concurrent. Let 
be the intersecting point of these three lines.
is tangent to the circumcircle of . Therefore 
, so lies on the radical axis of and the nine point circle of 
. lies on the polar of so lies on the polar of . So line 
is the polar of . So are concurrent at the point which is the pole of line 
. This line is the radical axis of and the nine point circle of , so it is perpendicular to the Euler line of . Since 
, it follows that is on the Euler line.
and be the circumcenter and orthocenter of . be the barycenter of 
as the intersection of the tangent at to 
and the line . By applying Pascal's Theorem to 
, we obtain that lies on the -midline of .
are collinear and that the pole of the line formed by these three points lies on 
, we proceed as follows:
, and let be the intersection of 
with . Define points 
and 
similarly. is the intersection of the tangents at and to . Furthermore, from this post (the part concerning cross-ratios), we deduce that 
concur at .
such that ![$$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$](http://latex.artofproblemsolving.com/d/2/4/d24b46e65a587fbd977467b6e4e31838a0eb57aa.png)
forall 
