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 
,
,
be non-negative real numbers, no two of which are zero. Prove that :
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
Let 
be a triangle inscribed in circle 
and 
a point in its interior. The lines 
and 
intersect circle for the second time at 
and 
respectively. If 
are the reflections of 
with respect to the lines 
respectively, prove that the triangles and 
are similar.
be a triangle inscribed in circle 
and 
a point in its interior. The lines 
and 
intersect circle for the second time at 
and 
respectively. If 
are the reflections of 
with respect to the lines 
respectively, prove that the triangles and 
are similar.
