AoPS Academy
+1 w
[/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 convex hexagon with 
and 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
, show that 
.
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
, with 
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say . Replace the number in one of the squares with 
and erase the number in the other square.
, while the other squares are empty. Determine, as a function of , the maximum number of turns Alex can make.
be a point in the plane, distinct from the vertices of 
. Consider 
the reflections of with respect to lines 
and 
, in this order.
are collinear if and only if lies on the circumcircle of . does not lie on the circumcircle of and the centroids of triangles and 
coincide, prove that is equilateral.
be a fixed square and 
a variable point on segment 
. The square 
is constructed such that 
is on segment 
and 
is on segment 
. Let 
be the intersection point of lines 
and 
. Find the locus of as varies along segment .
be a convex quadrilateral, and , 
, and be the midpoints of diagonals 
and 
, and side , respectively. Also, suppose that 
and that 
, 
. Calculate the perimeter of triangle 
.
be the circumcircle of triangle 
with center 
, and the 
inmixtilinear circle is tangent to 
at 
respectively. is the intersection of 
and 
and 
is the intersection of 
and . Prove that the isogonal conjugate of lies on the line passing through the midpoint of 
and .
be an acute triangle with 
, and let 
be the projection from onto . Let 
be a point on the extension of past such that 
. Let 
be on the perpendicular bisector of 
such that and are on the same side of and![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of across 
and be 
and 
, respectively. Let 
and 
such that 
. Let 
and 
intersect the circumcircles of 
and 
at 
and , respectively. Let 
and 
intersect 
and 
at and 
. Let 
intersect at 
. Prove that 
.
with circumcenter and orthocenter 
, let be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line . Let be the point on the circumcircle of triangle such that is perpendicular to , and let be the point on line such that 
. Let and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let be the midpoint of 
. and 
are perpendicular.
and 
are perpendicular.
. If 
and 
, show that the area of 
is not less than the area of 
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
, by the midpoint of , and by 
the orthocenter of . Let 
be the point of intersection of the circumcircle 
of the triangle and the half line 
, and 
be the point of intersection (other than ) of the line 
and the circle . Prove that 
must hold.
the length of the line segment .)
be a parallelogram. Let line 
externally bisect 
and let 
be the line passing through which is parallel to line . Suppose that meets line at point and at point , and that meets the internal bisector of 
at point . Further let circle 
meet line 
at point 
and the internal bisector of meet circle 
at point 
., , , and 
are concyclic. be an acute-angled triangle, with circumcenter , circumradius and orthocenter . Let 
be a point on such that 
. Define 
and 
similarly.
, prove that is equilateral.
is divisibly by 
.
and![\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/c/8/9/c895ae7fdf8d7f284ac9fc94cc077d6edad6cbf0.png)
.
be the least divisor of n.
