be positive real numbers such that 
.![\[\frac{1}{a^2+155}+\frac{1}{b^2+155}+\frac{1}{c^2+155}\le \frac{11}{780}\]](http://latex.artofproblemsolving.com/4/2/7/427e12cbfdbbba6f26407dbb5c2430aee26c15eb.png)
When does the equality hold ?
and 
Prove that
where 
,
,
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 Adventure10, Mango247, and 1 other user
We are given a circle 
and two points 
so that 
.The circle 
(
) crosses the circle 
at C,D (
belongs to the short arc 
).From 
we consider the tangent lines 
to the circle 
,in such way that 
lays out of the circle .If 
show that the quadrilateral 
is cyclic.
and two points 
so that 
.The circle 
(
) crosses the circle 
at C,D (
belongs to the short arc 
).From 
we consider the tangent lines 
to the circle 
,in such way that 
lays out of the circle .If 
show that the quadrilateral 
is cyclic.
intersect 
.
and 
touches 
.
,
collinear and applying Pascal on 
,
collinear. So, 
are all collinear. And 
is the diameter of 
.
we must have 
.
.
be 
,
implies-![\[B = c \cap c_2 \Rightarrow \text{ (inverse of } B ) = CD \cap EF\]](http://latex.artofproblemsolving.com/6/b/2/6b2651f84094ebf23128e15c9eb316d38128223f.png)
,
and inverse of 
.
:
