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 the roots of the polynomial 
. Find 
if within that number exists a digit 
that appears exactly 
times in that number, hence find the number of that consist of 4 digits (Numbers may contain a 0)
be the real roots of![\[x^3 + 7x^2 -6x - 13 = 0\]](http://latex.artofproblemsolving.com/6/6/3/6639be4c9e92b5ad4466359681db0d2a461ec8dd.png)
Find 
be a triangle with incenter 
. Let 
be the point of intersection between the incircle and the side 
, the points 
and 
are in the rays 
and 
, respectively, such that 
and 
. Prove that 
.
with 
-excenter 
, the foot of the perpendicular from to is . Let the midpoint of segment 
be 
, 
lies on arc (not containing ) satisfying 
, 
intersects the circumcircle of at 
. If 
and intersect at 
, the perpendicular bisector of 
intersects 
at 
respectively, prove that 
are concurrent.
, 
, 
, be four points such that no three are collinear and is not the orthocenter of . Let , , 
be the orthocenters of 
, 
, 
, respectively. Suppose that the lines 
, 
, 
are pairwise distinct and are concurrent. Show that the four points , , , lie on a circle.
and 
be the circumcenter and orthocenter, respectively, of an acute scalene triangle . The perpendicular bisector of 
intersects 
and 
at 
and 
respectively. Let 
denote the intersection of the circumcircles of triangles 
and 
other than .
and 
analogously by repeating this construction two more times. Prove that , , , and are concyclic.
be a quadrilateral inscribed in a circle 
. Let 
, 
, 
, 
, 
, 
, 
, 
. Prove that 
lies over 
.
is circumcircle of an acute triangle . is midpoint of arc 
and is incenter of triangle . Let 
intersect in 
and for second time in 
. Let be a point on line 
such that 
is parallel to 
. Prove that is bisector of angle 
.
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 . A point lies on segment , and lies on segment . Let ray 
. A point , which lies on , bisects and it is on the other side of with respect to . Ray 
, ray 
, and 
. Prove that 
are cyclic.
