,
,
with center 
.
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] $](http://latex.artofproblemsolving.com/2/1/b/21b329d0a92c18c6a2861742177969d5977d4558.png)
. Prove that
.
.
from the domain 
and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
Y by rightways, Adventure10
Given two circles 
and 
with centers 
and 
that intersect at the points 
and 
.Passes through A two lines that intersect the circle at the points 
and 
, and the circle at the points 
and 
(points 
in colinear). Denote the midpoints of the segments 
and ![$M_{1}M_{2]}$](//latex.artofproblemsolving.com/1/c/f/1cf6a9d6de85b5b07917e14424a8af4787f76e9d.png)
, through 
and 
.Prove that:
Points 
and lie on a circle
The center of the circle passing through and lies in the middle of the segment 
and 
with centers 
and 
that intersect at the points 
and 
.Passes through A two lines that intersect the circle at the points 
and 
, and the circle at the points 
and 
(points 
in colinear). Denote the midpoints of the segments 
and ![$M_{1}M_{2]}$](http://latex.artofproblemsolving.com/1/c/f/1cf6a9d6de85b5b07917e14424a8af4787f76e9d.png)
, through 
and 
.Prove that:
Points 
and lie on a circle
The center of the circle passing through and lies in the middle of the segment 
(angles) 
.
is same, when 
and 
the diametrically opposite points of 
.
is right-angled, so its circumcenter is the midpoint of 
,![$[O_1O_2]$](http://latex.artofproblemsolving.com/d/6/6/d663fe1660add9433a47fda5bf4e50963f63b7f9.png)
)
