AoPS Academy
, 
. The internal angle bisector of 
and the external angle bisector of intersect the ray 
at points 
and 
, respectively. Given that 
, prove that 
.
+1 w
be an acute triangle, and let 
and 
be its circumcenter and orthocenter, respectively. For 
, points 
and 
lie on lines 
and 
(where 
), respectively, such that 
is a parallelogram. Prove that![\[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]](http://latex.artofproblemsolving.com/b/a/5/ba563336236b6a04fc9ef661760fa1251bfb07d6.png)
in this order. Prove that![\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]](http://latex.artofproblemsolving.com/9/e/e/9ee73bbdc4b4f2cad2eb3fcfb3dbdf76b6200b4d.png)
be the center of circumcircle 
of acuted-triangle 
. Another circle 
with center 
is tangent internally to circle at point 
and side
at point . Also circle intersect the lines 
and 
respectively at points and 
, and lines 
and 
respectively at points 
and , 
and . Lines 
and 
intersect at
. Prove 
.
.
