AoPS Academy
opposite the vertex 
be tangent to the side 
at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
be the circumcircle of triangle , 
is the midpoint of and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be positive integers such that 
and 
. Find the maximum possible value of 
.
. 
are the midpoints of 
respectively. is the midpoint of arc 
, 
is symmetric to through 
. Let 
be the incenter and tangent of angles of triangle 
respectively. Prove that 
have a common point
is the least angle in 
. Point 
is on the arc from the circumcircle of . The perpendicular bisectors of the segments 
intersect the line 
at 
, respectively. Point 
is the meet point of 
. Suppose that 
is the radius of the circumcircle of . Prove that:![\[ BT+CT\leq{2R}. \]](http://latex.artofproblemsolving.com/7/9/3/793c0bdc27571a1b063a861cee81bfba297b0738.png)
is the least angle in . Point is on the arc from the circumcircle of . The perpendicular bisectors of the segments intersect the line at , respectively. Point is the meet point of . Suppose that is the radius of the circumcircle of . Prove that:
![\[\angle{BTC}=2A\]](http://latex.artofproblemsolving.com/f/f/3/ff36a26f21e3409b7407bd1111811c108f8b4cb5.png)
, ![\[\dfrac{BC}{\sin{A}}=2R\]](http://latex.artofproblemsolving.com/7/b/9/7b9bca9ba579d222e09fd803629fdfa228c72da1.png)
, ![\[\dfrac{\sin{2A}}{BC}=\dfrac{\sin{x} +\sin{y}}{BT+CT}\]](http://latex.artofproblemsolving.com/9/e/a/9ea956289a98449cfde021fa1fe8167316d01ee3.png)
![\[\sin{x}+\sin{y}\le2\sin{\dfrac{x+y}{2}}=2\cos{A}\]](http://latex.artofproblemsolving.com/3/d/0/3d02c860214736bcc6067bafa20887d5145bae03.png)
![\[\dfrac{\sin{2A}}{BC}\le\dfrac{2\cos{A}}{BT+CT}\]](http://latex.artofproblemsolving.com/f/5/2/f5229cd294c3a27af030c56a0def74f65efc5f6b.png)
![\[\Rightarrow BT+CT\le\dfrac{BC}{\sin{A}}=2R\]](http://latex.artofproblemsolving.com/a/b/d/abd819e0adbde17b478e771c041fdb0fb7a03bc9.png)
and 
.
specifies the greatest common divisor of 
and 
.
be real numbers so that 
and 
Show that
and 
such that:
.
,
is an isosceles trapezoid and 
,
.
.
as 
and 
is achieved iff 
,