AoPS Academy
be a triangle and 
its circumcentre. A line tangent to the circumcircle of the triangle 
intersects sides 
at 
and 
at 
. Let 
be the image of 
under 
. Prove that the circumcircle of the triangle 
is tangent to the circumcircle of triangle .
be a semicircle with diameter and midpoint 
. Let 
be a point on different from and 
.
touches in a point 
, the segment 
in a point , and additionally the segment 
. The circle 
touches in a point and additionally the segments 
and .
and 
are perpendicular.
with no parallel sides is inscribed in a circle 
. Circles 
are inscribed in triangles 
, respectively. Common external tangents are drawn between 
and 
, and 
, and 
, and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle 
. Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.
that are defined and continuous on the interval 
and that satisfy the functional equation![\[f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).\]](http://latex.artofproblemsolving.com/c/8/d/c8dc7392cab8b1f10c9df1c31e98215d3baa7f60.png)
that are defined and continuous on the interval and that satisfy the functional equation
: 
is continuous and the equation becomes 
with 
plus the fact that 
implies 
to conclude 
for any 
such that 
in order to have continuity.
