be an isosceles triangle with 
Let 
be a point on the circumcircle of 
Let 
intersect the extension of 
at 
Let 
be the midpoint of segment 
and let 
be the intersection of 
and 
Let the extension of 
intersect 
and the circumcircle of 
and 
respectively. Given that 
and 
compute the length of 
.
be a subset of 
such that no sum of three distinct elements of 
.
.
that satisfy the equation 
.
be a circle, and let a line 
intersect 
and 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
Y by
Triangle 
is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle .
is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle .This post has been edited 2 times. Last edited by parmenides51, Dec 22, 2022, 12:42 AM
.![$[AFY]=x, [CFX]=y$](http://latex.artofproblemsolving.com/8/0/b/80b2862712c328205c751ab661bff0baedcb5591.png)
.
and 
,
.
.
