,
,
with center 
.
the midpoint of side 
intersects the circle 
.
be a point on the circle 
and 
.
.
satisfying 
Prove that 
When does equality hold?
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
and 
.
![$ a,b,c\in [0,1] $](http://latex.artofproblemsolving.com/2/1/b/21b329d0a92c18c6a2861742177969d5977d4558.png)
. Prove that
.
.
from the domain 
and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
Y by doxuanlong15052000, Adventure10
Given triangle 
and let 
be the foot of angle bisectors of 
,respectively.
lie on 
such that 
. Let 
be the foot of 
-altitude on 
.
Points
lie on such that triangles 
are correspondingly similiar (with the given order of vertices) such that 
and 
.
Show that:
and let 
be the foot of angle bisectors of 
,respectively.
lie on 
such that 
. Let 
be the foot of 
-altitude on 
.Points
lie on such that triangles 
are correspondingly similiar (with the given order of vertices) such that 
and 
.Show that:
let 
the proyections of 
at 
a point at 
such that 
is tangent to 
,
is cyclical 
and 
and 
points at 
and 
and 
is evident that 
foot of 
with 
and 
intersection of 
with 
the point on 
and 
reflection of 
about 
.
intersection of 
we can see that 
so 
are cyclic .
and we're done 
