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.
, the point 
is the foot of the altitude from 
, and 
is a point on the segment 
. The lines through parallel to 
and 
meet 
at 
and 
, respectively. Points 
and 
lie on the circles 
and 
, respectively, such that 
and 
.
and 
are concyclic.
(i.e., the square on 
has as one of its sides and lies outside ). Show that the lines drawn from the vertices 
, 
, and 
to the centers of the opposite squares are concurrent.
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
. Find all possible values of 
be the solid square region with vertices at![\[
(0, 0), \left(\frac{1}{2}, \frac{1}{2}\right), (1, 0), \left(\frac{1}{2}, -\frac{1}{2}\right).
\]](http://latex.artofproblemsolving.com/a/9/8/a9898f9237f07eaef5d17a9ac689d80ca631863f.png)
consists of two copies of 
: one copy which is rotated 
counterclockwise around the origin and scaled by a factor of 
, and another copy which is also rotated counterclockwise around the origin and scaled by a factor of , and then translated by 
. and 
below.
be the limiting region of the sequence 
. can be written as 
for relatively prime positive integers 
and 
. Find 
.
be an acute triangle and let 
be the midpoint of . A circle 
passing through 
and meets the sides and at points and 
respectively. Let 
be the point such that 
is a parallelogram. Suppose that lies on the circumcircle of . Determine all possible values of 
.
be a triangle with circumcircle 
. Let 
and 
respectively denote the midpoints of the arcs and that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc including ). Let 
be the incenter of . Let 
be the circle that is tangent to and internally tangent to at , and let 
be the circle that is tangent to and internally tangent to at . Show that the line 
, and the lines through the intersections of and , meet on .
be a circle with centre , and 
a convex quadrilateral such that each of the segments 
and 
is tangent to . Let be the circumcircle of the triangle 
. The extension of 
beyond meets at 
, and the extension of 
beyond 
meets at 
. The extensions of 
and 
beyond meet at and , respectively. Prove that ![\[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]](http://latex.artofproblemsolving.com/8/2/6/8261276053071cc6cc06dda824a883426bf1d4ac.png)
such that for every 
is satisfied? Explain the answer.
be a random point on the smaller arc of the circumcircle of square 
, and let 
be the intersection point of segments and 
. The feet of the tangents from point to the circumcircle of the triangle 
are and , where 
is the center of the square. Prove that points , , and lie on a single circle.
and 
such that 
cannot completely cover a square of side length 
, if the three smaller squares do not overlap in their interiors (i.e., they may touch at edges or corners, but no part of one lies over another).
,
for all 
,
,
.
.
