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 in the interior of . The following ratio equalities hold:![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
.
with 
. The circumcircle 
of has radius 
. There is a point in the interior of the line segment such that 
and the length of 
is . The perpendicular bisector of 
intersects at the points 
and 
. is the incentre of triangle 
.
midpoint of side 
in an isosceles triangle 
with 
. Take a point 
on a smaller arc 
of circumcircle of triangle 
. Denote by 
point inside of angle 
such that 
and 
.
does not depend on choice of . be an acute triangle (
) with circumcircle 
. Point 
is chosen on arc 
, not containing 
, so that, letting 
intersect at 
, one has 
. Points and lie on lines and 
respectively, such that 
is a parallelogram. Point is chosen on arc , not containing 
, such that 
. Line 
intersects 
at 
, and line 
intersects 
at 
. Line 
intersects , and 
at points 
, 
and 
respectively.
.
be the incircle of the triangle and with centre 
. Let 
be the circumcircle of the triangle 
. Circles and 
intersect at the point and . Let 
be the intersection of the common tangents of the circles and . Show that the circumcircle of the triangle 
is tangent to the circumcircle of the triangle .
, 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.
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
. Find all possible values of 
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 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)
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.
, 
, is the point on such that 
. Let 
be the circle through tangent to 
at , and 
the circle through tangent to at . Let 
be the second intersection of and . Prove that lies on .
be a point on ray beyond is tangent to circle 
and is tangent to circle 
is cyclic.
, giving the required collinearity.
be the intersection of and . We will prove that 
, which implies![\[
(!) \quad AB \text{ is tangent to } (CF'B).
\]](http://latex.artofproblemsolving.com/0/9/1/0912a5d89c9b96991df593e9811fe2ccb3ad79db.png)
Due to the Power of a Point theorem,![\[
AD^2 = AF \cdot AC.
\]](http://latex.artofproblemsolving.com/c/7/c/c7ce7f6916003527f4d1812c1cfcaf61d3bf0bf6.png)
Since 
, we have![\[
AB^2 = AD^2 = AF \cdot AC.
\]](http://latex.artofproblemsolving.com/e/f/f/eff6902bf20790e7dfdb273eee89ca8dac4c58c8.png)
. Note that 
is the radical axis of 
and 
, and so this implies that lies on extended, which then implies lies on .
.
.
.
such that 
.![$$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$](http://latex.artofproblemsolving.com/f/0/7/f0731fbe5c83aedc8474975f930f7194c08cc070.png)
be 
,
is tangent to 
and 
meet on